Research Statements for the Second Young Geometric Group Theory Meeting at the Center for Mathematical Sciences, Technion, Haifa, Israel, February 2013

Edited by Tobias Hartnick, Sang Rae Lee, Vera Toni´c Contents

Chapter 1. Combinatorial group theory, free groups, groups acting on trees 1 Chapter 2. Metric aspects of ﬁnitely-generated groups 5 Chapter 3. (Co-)Homological properties and ﬁnite properties of groups 13

Chapter 4. Analytic, ergodic and probabilistic methods in group theory 21 Chapter 5. Systolic and (relatively) Gromov-hyperbolic groups 29

Chapter 6. Surface groups, mapping class groups and Out(Fn) 37 Chapter 7. Three manifold groups, knots and braids 43

Chapter 8. Lattices in locally compact groups 47 Chapter 9. Buildings, Coxeter groups and Artin groups 51 Chapter 10. Groups acting on cubical complexes and spaces with walls 55

Chapter 11. Bounded cohomology, simplicial volume and quasimorphisms 65 Chapter 12. Locally-compact groups, diﬀeomorphism groups, other non-discrete groups 73 Chapter 13. Other aspects of geometric group theory: Algebraic geometry, Thompsons groups, tree manifolds 77

Alphabetical List of Contributors 81

v Abstract

This booklet contains the research statements submitted by the participants of the Second Young Geometric Group Theory Meeting, grouped together by research areas. We would like to thank the Center of Mathematical Sciences, the Israeli Science Foundation, European Research Council and the US National Science Foundation for ﬁnancial support.

Editors’ Remark: We have grouped the research statements by subjects. In case of multiple subjects, we have decided on a primary subject, where the research statement is printed, and secondary subjects, where the research statement is merely linked to. A complete alphabetical list of contributors is given at the very end of this booklet.

Received by the editor February 2, 2013. Key words and phrases. Geometric group theory.

vi CHAPTER 1

Combinatorial group theory, free groups, groups acting on trees

Ayala Byron My main research interest has to do with possible links between geometric group theory and logic. In his paper ”Diophantine geometry over groups. I, Makanin- Razborov diagrams” Sela introduced Limit groups and used their JSJ-decomposition to study the structure of deﬁnable sets in the free group. Another important geometric tool developed in this work is the notion of a Hyperbolic Tower. A group G is a hyperbolic tower over H ≤ G if there is a sequence

H ≤ G0 ≤ G1 ≤ ... ≤ Gk = G such that:

• G0 = H ∗F ∗S1 ∗...∗Sp, where F is a free group (and can be trivial), p ≥ 0 and the Si ’s are the fundamental groups of surfaces of Euler characteristic ≤ −2. • For each 1 ≤ j ≤ k, Gj has a non-trivial JSJ-decomposition, where one of the vertex groups is Gj−1, the rest are surface groups which are “attached” to Gj−1 in a way that follows certain conditions, and there is a retract rj : Gj → Gj−1 under which the images of these surface groups are non-abelian. A first order formula in the language of groups (with constants in the free group on n generators, Fn) is a finite formula which uses variables, the basic logic symbols, elements of the free group as constants and quantifiers. A formula in which one of the variables is not quantified defines a subset of the free group - the set of elements which, when taken to be the value of the non-quantified variable, make the formula a true sentence in the free group. More generaly, a formula with k non-quantified k variables defines a subset of Fn . These are the definable sets in Fn. We hope to use these tools to study the possible algebraic structures of definable sets in Fn.

Aglaia Myropolska

Let G be a ﬁnitely-generated group and n ≥ rank(G). Consider Γn(G) to be the set of all generating systems (g1, ..., gn) of G. We deﬁne a structure of graphs on −1 Γn(G) by connecting (g1, ..., gn) to (g1, ..., gigj, ..., gn), i 6= j and (g1, ..., gi , ..., gn). Γn usually is called the product replacement graph. The transformations above are called elementary Nielsen moves. They generate the automorphism group of the free group. Therefore, every connected component of Γn(G) is a Schreier graph of AutFn.

1 2 1. COMBINATORIAL GROUP THEORY

We consider infinite finitely-generated groups and study the geometric behaviour of these graphs such as: - connectedness. It is equivalent to a classical question about transitivity of the action of Aut(Fn) on the set of generating systems of a given group. There are results on connectedness of, for instance, nilpotent groups [2] and finite solvable groups [1]. However the connectedness of Γn(S), n ≥ 3 for finite simple groups is still open (Wiegold conjecture). We study connectedness of basic examples of self-similar groups (such as Grigorchuk groups or the Gupta-Sidki group). − (non)amenability. As an analogue of ’non-amenability’ in the world of finite graphs, {Γn(Gk)}n≥rank(Gk) is a family of expanders for a family of finite abelian groups {Gk} [5]. Gambord and Pak [3] proved that the statement is true for the family {PSL(2, p)}p→∞ [3]. As an example of infinite group we take free group. Then the Schreier graph becomes the Cayley graph of Aut(Fn) which is not amenable. It is not evident though if for any infinite finitely-generated group the corresponding Schreier graphs of Aut(Fn) are non-amenable [4]. References [1] M.J. Dunwoody, Nielsen transformations, in: Computation Problems in Ab- stract Algebra, Pergamon, Oxford, 1970, 45-46. [2] M.J.Evans, Presentations of groups, involving more generators than are necessary, Proc.London Math.Soc. (3) 67 (1993), n.1, 106-126. [3] A.Gamburd, I.Pak, Expansion of product replacement graphs, Combinatorica 26 (2006), n.4, 411-429. [4] R.I.Grigorchuk and V.V.Nekrashevych, Amenable actions of non-amenable groups, preprint. [5] A. Lubotzky, I. Pak, The product replacement algorithm and Kazhdan’s property (T), Journal of the AMS 14 (2001), 347-363.

Ori Parzanchevski My current research is the study of expansion properties in simplicial complexes. Using tools from topology and geometry, I study the extent to which phenomena from the theory of expander graphs have parallel in higher dimension. This include, for example, isoperimetric inequalities, quasi-randomness, the behavior of random walk, and geometric expansion a la Gromov. In addition, I have been studying word maps in groups. In particular, the Fourier expansion related with a word map, and the question whether words inducing identical distribution are automorphism-equivalent.

Doron Puder The theme that connects all my lines of research are formal words (elements of free groups) and in particular primitive elements, that is words which belong to some free generating set. Associated with w ∈ Fk and any group G is the word map w : G × ... × G → G defined on the direct product of k copies of G. For compact G, one can take the Haar measure on G × ... × G (for finite groups this is simply the uniform distribution) and then push it forward via the word map w to obtain some measure on G. It is an easy observation that if w1 and w2 belong to the same orbit of the Aut (Fk)-action on Fk, then they induce the same measure on every compact group G. One question I’m considering is weather the converse also holds. Namely, given RIZOS SKLINOS 3 two words w1, w2 ∈ Fk from different Aut (Fk)-orbits, is it always possible to come up with a compact group on which they induce different measure? Together with a fellow student, O. Parzanchevski, we have answered this to the positive in the special case where w1 is primitive [1]. The proof relies heavily on new techniques with Stallings Core graphs. With further work, these results about primitive words have also some consequences regarding expansion properties of random graphs. Another question I’m interested in is the following: given a representation ρ : Fk → GLn (C), to what extent do the character values of the primitive elements determine ρ? References [1] Doron Puder and Ori Parzancheski. Measure Preserving Words are Primitive. Submitted.

Rizos Sklinos My research is focused on model theory and its interactions with geometric group theory. Much attention has been given to the subject after Sela [6] and Kharlampovich- Myasnikov [1] independently proved that non abelian free groups share the same common theory. Furthermore, Sela [5] proved, using geometric methods, that this common theory is stable. In joint work with Perin [4] we give an algebraic description of forking in standard models of the theory of free groups. So, our result is: two tuples of elements of a free group are independent over the empty set if and only if they belong to associated free factors, i.e. free factors whose bases together extend to a basis of the whole group. We also give a description of forking over standard models, and over “big” parameter sets, i.e. sets over which the free group under consideration is freely indecomposable. It remains open to generalize the description over arbitrary parameter sets. In a joint work with Louder and Perin [2], we use a geometric tool introduced by Sela (Hyperbolic Tower) to prove that there exists a (finitely generated) model of the theory of free groups, in which we can find two maximal independent sequences of realizations of the generic type of different cardinalities. In the same paper we prove that the free product of two homogeneous groups is not necessarily a homogeneous group, answering a question of Jaligot. Moreover, in a recent work with Perin, Pillay and Tent [3] we use the elimination of imaginaries in torsion-free hyperbolic groups (as proved by Sela) in order to show that any (non-cyclic) torsion-free hyperbolic group is definably simple. In the same paper we prove that the generic type of a non-cyclic torsion-free hyperbolic group is foreign to any interpretable abelian group. The full conjecture being that there are no infinite interpretable fields in any model of the free group (or more generally in any model of any theory of a (non-cyclic) torsion-free hyperbolic group). References [1] Olga Kharlampovich and Alexei Myasnikov, Elementary theory of free non- abelian groups. J. Algebra, 302:451–552, 2006. [2] L. Louder, C. Perin, and R. Sklinos, Hyperbolic Towers and Independent Generic Sets in the Theory of Free Groups. To appear in the Proceedings of the conference Recent developments in Model theory, Notre Dame Journal of Formal Logic, 2012. [3] C. Perin, A. Pillay, R. Sklinos, and K. Tent, On Groups and Fields Interpretable in Torsion-Free Hyperbolic Groups. In preparation, 2012. 4 1. COMBINATORIAL GROUP THEORY

[4] C. Perin and R. Sklinos, Forking and JSJ decompositions in the Free Group. In preparation, 2012. [5] Zlil Sela, Diophantine geometry over groups VIII: Stability. To appear in the Ann. of Math. (2), available at http://www.ma.huji.ac.il/ zlil/. [6] Zlil Sela, Diophantine geometry over groups VI: The elementary theory of free groups, Geom. Funct. Anal., 16:707–730, 2006.

Carrie A. Whittle We are working on proving that the word problem in the automorphism groups of right-angled Artin groups is in P.

We construct a polynomial-time algorithm for the word problem in the automorphism groups of right-angled Artin groups. Our techniques generalize those of Schleimer where he ﬁnds a polynomial-time solution to the word problem for automorphism groups of free groups. CHAPTER 2

Metric aspects of ﬁnitely-generated groups

Khek Lun Harold Chao On the boundary of a CAT(0) space one can define the cone topology and the Tits metric, but the Tits metric is only lower semi-continuous with respect to the cone topology. For a general CAT(0) space, the cone topology of the boundary is not enough to determine the Tits metric. When a CAT(0) space admits a geometric group action, there are fewer candidates of possible Tits metrics on the boundary. Ruane has obtained rigidity results on the possible Tits metrics for two cases of one- dimensional boundaries, namely when the boundary is homeomorphic to a circle, or to a suspension of a Cantor set. In the first case the space is either R2 or else quasi-isometric to H2, and in the second case it splits as the product R × Y where Y is CAT(0) with boundary homeomorphic to a Cantor set. (Assume geodesic completeness here, see [1].) If the boundary of a CAT(0) space is zero-dimensional, and the space admits a cocompact group action, then the possible boundaries are either a two-point set or the Cantor set. Thus, for a CAT(0) space which could be a product of two CAT(0) spaces with zero-dimensional boundaries, its boundary can only be a circle, a suspension of a Cantor set, or a join of two Cantor sets. My current project, suggested and guided by my advisor Chris Connell, is to investigate in the remaining case of a join of two Cantor sets, to deduce a rigidity result for the Tits metric of the boundary, and to obtain a decomposition result for the space itself, analogous to the cases in Ruane’s paper. Along these lines, I proved the following results: If a CAT(0) space X has a boundary ∂X homeomorphic to the join of two Cantor sets, C1 and C2, and if X admits a geometric group action by a group G < Isom(X) 2 containing a subgroup isomorphic to Z , then its Tits boundary ∂TX is isometric to the spherical join of two uncountable discrete sets. Hence when X is geodesically complete, then X = X1 × X2 with ∂Xi homeomorphic to Ci, i = 1, 2, and either G or a subgroup of G of index 2 is a uniform lattice in Isom(X1) × Isom(X2). Furthermore, a finite index subgroup of G is a lattice in Isom(T1) × Isom(T2), where Ti is a bounded valence bushy tree quasi-isometric to Xi, i = 1, 2. References [1] Kim E. Ruane. CAT(0) groups with specified boundary. Algebraic and Geomet- ric Topology,6:633–649, 2006.

Adrien Le Boudec My main research interest is the study of asymptotic cones of Lie groups, as well as asymptotic cones of their lattices.

5 6 2. METRIC ASPECTS

The idea of asymptotic cone of a metric space was first used by Gromov in the proof of his polynomial growth theorem, and then a general definition was given by van den Dries and Wilkie. An asymptotic cone is a sort of picture of our space seen from infinitely far away, and contains information about large scale geometric properties of our original space. For instance, any asymptotic cone of a hyperbolic group, endowed with some word metric, is a real tree, and this easy fact admits a partial converse due to Gromov: if all the asymptotic cones of a finitely generated group are real trees, then the group is hyperbolic. One of the main question I am focusing on is the study of the existence of cut-points in asymptotic cones. Recall that a cut-point in a connected metric space is a point such that we get a disconnected space when we remove it. Examples of groups with cut-points in all their asymptotic cones include relatively hyperbolic groups by the work of Osin and Sapir, and mapping class groups by the work of Behrstock. In the opposite direction, the direct product of two infinite groups is easily seen to have no cut-point in any asymptotic cone, and non-virtually cyclic groups satisfying a law also give us a large class of groups without asymptotic cut-point. Drutu, Mozes and Sapir established a link between the existence of cut-points in asymptotic cones and the growth rate of the divergence function of a metric space, which roughly speaking estimates the cost of joining two given points while remaining outside a large ball. More precisely, they show that a finitely generated group has a linear divergence if and only if none of its asymptotic cones has cut-points. In particular they deduce that the asymptotic cones of SLn(OS ) do not have cut-points, where OS denotes the ring of S-integers of a number field. I proved that the same result holds in positive characteristic. The goal of my present research is to deduce properties of a space X, belonging to a particular class of metric spaces, from the fact that one asymptotic cone of X has cut-points.

Curtis Kent An asymptotic cone of a group G is the space obtained by observing G with a fixed word metric from infinitely far away. Or more precisely, an asymptotic cone of a group G with word metric, dS, is the ω-limit of the sequence of metric spaces (G, dS/ρn) where (ρn) is a divergent sequence of scaling constants and ω is an ul- trafilter.

Gromov asked whether the fundamental group of an asymptotic cone of a finitely generated group was always trivial or uncountable. It turns out that an asymptotic cone of a finitely generated group can have countable non-trivial fundamental group (due to Olshanskii, Osin, and Sapir) or even finite non-trivial fundamental groups (due to Cornulier and Tessera). I have shown that Gromov’s dichotomy does hold for many groups, including HNN extensions and amalgamated products where the associated subgroups are quasi-isometrically embedded prairie groups [3]. A prairie group is a group all of whose asymptotic cones are simply connected. Tim Riley in [4] proved that the methods for understanding the fundamental group of asymptotic cone can be generalized to the setting of higher homotopy groups which leads to the following question. Does Gromov’s dichotomy hold for higher homotopy groups, i.e. is πn of an asymptotic cone of a group always trivial or uncountable? FRANCESCO MATUCCI 7

I have shown that every continuous path in the asymptotic cone of a group is the limit of paths into the Cayley graph. The same holds true for discs when considering the Cayley complex. In [1,3] this is used to understand the fundamental group and local connectivity properties of the asymptotic cones for some well known groups. The methods used in [1,3] will also apply to understanding the local n-connectedness properties of asymptotic cones. References [1] G. Conner and C. Kent. Local topological properties of asymptotic cones of groups. Submitted to Algebraic & Geometric Topology [2] M. Gromov. Asymptotic invariants of infnite groups. In Geometric group theory, Vol. 2 (Sussex, 1991), pages 1-295. Cambridge Univ. Press, Cambridge, 1993. [3] C. Kent. Asymptotic cones of HNN extensions and amalgamated products. Sub- mitted to Algebraic & Geometric Topology [4] T.R. Riley. Higher connectedness of asymptotic cones. Topology, 42(6):1289- 1352, 2003.

Francesco Matucci Growth functions in groups have been widely studied and provide tools to dis- criminate groups. In his celebrated 1981 theorem, Gromov showed how the word growth function (which counts the number of elements in the n-th ball of a finitely generated group) describes completely finitely generated virtually nilpotent groups as those with growth function bounded above by a polynomial. A similar result has been obtained in 1993 by Lubotzky, Mann and Segal (see [5]) which classified residually finite virtually solvable groups of finite rank using the subgroup growth function (which counts all finite index subgroups of at most at most a given index). In 2007 Bou-Rabee [3] has introduced the Farb growth function for finitely generated residually groups which attempts to quantify the residual finiteness of a group in an efficient way. More precisely, given a finitely generated group G = hSi and for g ∈ G we define the quantity kG(g) = min{[G : N]: g 6∈ N and N is normal in G}. We define the Farb growth function as S RG(n) = max kG(g). g∈B(n) S It can be shown that the function RG(n) is essentially independent of the generating set S. This function has been studied for several groups and it has been shown by Bou-Rabee and McReynolds [2] that, for the case of linear groups, it gives a Gromov-like characterization for virtually nilpotent groups. In joint work together with M. Kassabov [4] we worked on the case of the free group by rephrasing the study of the growth in terms of laws in groups. Moreover, we established a connection of the Farb growth to a second growth called intersection growth which allows one to obtain information about the Farb growth. The intersection growth function iG(n) is defined to be the index of the the intersection of all subgroups of index at most n inside the finitely generated group G and can be seen as a variant of the subgroup growth function, but in many cases it is better behaved. Together with I. Biringer, K. Bou-Rabee and M.Kassabov [1] we have computed this function for several groups and are in the course finding parallels to Grunewald, Lubotzky, Segal and Smith’s work on the relation between the subgroup growth and the nilpotency of a group (see [5]). 8 2. METRIC ASPECTS

References [1] I. Biringer, K. Bou-Rabee, M. Kassabov, F. Matucci Intersection growth in groups. in preparation. [2] K. Bou-Rabee and B. McReynolds. Asymptotic growth and least common mul- tiples in groups. submitted. [3] Khalid Bou-Rabee. Quantifying residual finiteness. J. Algebra, 323(3):729–737, 2010. [4] Martin Kassabov and Francesco Matucci. Bounding the residual finiteness of free groups. Proc. Amer. Math. Soc., 139(7):2281–2286, 2011. [5] Alexander Lubotzky and Dan Segal. Subgroup growth, volume 212 of Progress in Mathematics. BirkhäuserVerlag, Basel, 2003.

Thang Nguyen I am interested mainly in rigidity problems in geometry and geometric group theory. In particular, I am interested in the following questions: • Quasi-isometries between spaces like symmetric spaces, buildings, cube complexes,... and quasi-isometric embedding into other spaces like product of trees or L1-functions. • relationship between vanishing of cohomologies and the rigidity of isometric actions on certain CAT(0)-spaces. • quasi-isometric rigidity problems. I have got some partial results about quasi-isometric embedding problems, have been thinking about the second problem, and I am looking for a good start on the third one. Concerning the ﬁrst problem, think about bi-Lipschitz embedding instead of quasi-isometric embedding (this is automatic if the spaces are discrete, and needs a little modiﬁcation if they are not discrete). The induced metric from the embedding is a conditionally of negative type kernel, therefore property (T) of higher rank semisimple groups prevents any invariant embeddings (the induced metric is invariant) or even embeddings with small distortions from symmetric spaces into L1. This leads to an interesting question: we know the groups Sp(1, n) can be bi-Lipschitz embedded into L1, but it has property (T), so the embedding has to be distorted. How small can the distortion be and does it have anything to do with the Kazhdan constant of the respective group?

Concerning the question of rigidity of actions on CAT(0)-space, I am interested in the following question: What is the relation between harmonic functions on the target CAT(0)-spaces and cohomologies with respect to representations arising from the action? Concerning the third question, I am looking for an appropriate problem to start with. I am particularly interested in this question in the context of actions on the circle or on spheres.

Yaar Solomon My main interest in mathematics is on questions that has either geometric or combinatorial nature. During the last several years, most of my research dealt with questions on tilings, and inﬁnite discrete subsets of Euclidean spaces. I studied the connection between these two objects, and answered questions concerning the geometrical properties of them. ALEXEY TALAMBUTSA 9

A set Y ⊆ Rd is called a separated net (or a Delone set) if there exist constants d r, R > 0 such that for any y1, y2 ∈ Y we have d(y1, y2) ≥ r, and for every x ∈ R d we have d(x, Y ) ≤ R. For separated nets Y1,Y2 ⊆ R we say that Y1 is biLipschitz −1 to Y2 if there is a bijection ϕ : Y1 → Y2 such that ϕ and ϕ are both Lipschitz maps. Y1 is a bounded displacement of Y2 (BD) if there exists a constant α > 0 and a bijection ϕ : Y1 → α · Y2 such that sup {d(y, ϕ(y))} < ∞. y∈Y1 The following question was posed by Furstenberg, already in the sixties, and again by Gromov in 93: Is every separated net biLipschitz equivalent to a lattice? The motivation for this question came from Geometric Group Theory, since two metric spaces are quasi-isometric if and only if they contain biLipschitz equivalent separated nets. This question was answered negatively (for d > 1) by Burago and Kleiner, and independently by McMullen. But given a speciﬁc separated net Y , like the vertices of a Penrose tiling for example, it is not trivial to say whether Y is biLipschitz to a lattice or not. I studied the biLipschitz equivalent classes, and also the BD equivalent classes, of separated nets arising from a special class of non-periodic tilings of Rd - substitution tilings. I showed that such separated nets are always biLipschitz to a lattice, and found a tight condition on their related eigenvalues saying when the net is a BD of a lattice. In my current research I am interested in related questions in geometric group theory, and also in other new questions on these objects. Here are some examples: (1) In light of the discussion above, McMullen and Burago-Kleiner actually shows that biLipschitz equivalence and bijective quasi-isometry equivalence is not the same for separated nets in Rd, d > 1 (any two separated nets are bijectively quasi-isometric). This distinction is not known for many ﬁnitely generated groups, equipped with the word metric. In particular, the following special case is open:

Question Given a ﬁnitely generated nilpotent group G, is it biLipschitz equivalent to any ﬁnite index subgroup of it? (2) Another interesting question regarding separated nets is the following. A set X ⊆ Rd is a Danzer set if it intersects every convex set of volume 1.

Question Is there a Danzer set in Rd (d > 1) which is also a separated net?

Alexey Talambutsa My current research interests are in combinatorial and geometric group theory, especially growth functions of the groups, introduced by Milnor and Svarc. If S is a finite set of generators for the group G, then the growth function fG,S(n) is the number of group elements that can be expressed as words of length less or equal then n in the alphabet S ∪ S−1. From a geometric viewpoint this function counts the number of points in the balls of radius n in the Cayley graph Cay(G, S) supplied with usual discrete metric. While several remarkable results by Bass–Guivarch–Gromov, Grigorchuk and J.Wilson gave beautiful insights about the hierarchy of the growth functions, there are still many interesting problems to be solved. My former results were related to the study 10 2. METRIC ASPECTS of asymptotic behaviour of fG,S(n) and its dependence on the set S. More pre- pn cisely, they were related to the study of the limit λ(G, S) = limn→∞ fG,S(n) that is called an exponential growth rate of the group G for the generating set S. Even for some ”simple” groups it can be hard to compute λ(G, S), for example finding this constant for the Baumslag-Solitar group BS(2, 3) = ha, t | ta2t−1 = a3i seems to be a real challenge. And even when it is possible to compute the value λ(G, S) for some generating set S, it can be quite complicated to prove that this value is equal to λ(G) = inf{λ(G, S0)} (where S0 runs along all finite sets that generate G). In my former works could establish some lower and upper bounds for λ(G) for some classical groups like some one-relator groups, Hecke groups. In a recent joint work with Michelle Bucher we could prove some gap property for the growth rates of free products and amalgamated free products, and shown that the minimal value is established for the classical group PGL(2,Z). One of interesting question here is the min- imality of the λ(π1(Sg),S) for the standard generating set S = {s1, t1, . . . , sg, tg} of the fundamental group π1(Sg) = hs1, t1, . . . , sg, tg | [s1, t1] ... [sg, tg] = 1i of the closed orientable surface of genus g ≥ 2. Another exciting problem related to growth which attracts me a lot is the question whether exponentiality of the growth necessarily forces the Cayley graph of the group to have isometrically embedded 3-trees (for non-amenable groups it was proved by Benjamini and Schramm, for the solvable groups - by de Cornulier and Tessera). Besides the discussed questions of growth I am quite interested in decision problems related to group and semigroup theory and also in the questions of computational complexity for the problems that arise in this field. The question that I aim at here is the complexity of quadratic equations for finite groups. While the complexity of general equations in most of non-abelian groups is known to lie inside the class of NP-complete problems (due to Goldmann and A.Russell), it is not clear whether the same would hold for quadratic equations.

Gasper Zadnik My research is in the area of CAT(0) spaces. My degree (2010) is the review of geometry of CAT(0) spaces which admit some geometric (i.e. proper cocompact isometric) group action and algebraic properties of groups which act so. I still follow the spirit of the degree. At the beginning of my PhD-studies I studied Riemannian symmetric spaces SL(n, R)/O(n) from the viewpoint of CAT(0) geometry and answered a question about the classification of nonidentity components of isometries of SL(3, R)/O(3) posed by K. Fujiwara in his talk, see [4, Problem 4.1]. Now I am focussing on the flat closing problem, the conjectural converse of the flat torus theorem. It asks whether a group G acting geometrically on a CAT(0) space X contains Z × Z, if X contains isometrically embeded copy of R × R (see [5] for example). In joint work with prof. Pierre-Emmanuel Caprace, we give a very partial answer to this question in the case when X is geodesically complete reducible CAT(0) space, see [3]. Our approach highly uses the structure theory of the whole isometry group of X, see [2]. Another special example of this problem is the following:

Conjecture. Let G be a group acting geometrically on a locally compact CAT(0) space X. Assume that all the elements of G (except the identity) act as a rank one GASPER ZADNIK 11 isometry. Then G is a hyperbolic group (in the sense of M. Gromov).

One possible approach to the problem is convergence action, namely to prove that the induced action of G on the ideal boundary of X is a convergence action and than try to deduce assumptions from Bowditch’s theorem in [1] which ensures the hyperbolicity of G. There are also several greater problems which I am interested in, let me mention some of them: • duality condition for CAT(0) groups (does for every geodesic line c in

CAT(0) space X there exists a sequence of elements (gn)n∈N from a group −1 G acting geometrically on X such that gnx tends to one and gn x to another endpoint of c in ∂X for some (any) x ∈ X); • are π and ∞ the only posible radii of Titz boundary of CAT(0) spaces, admitting some geometric group action (or, more general, action with full limit set at the ideal boundary)? • Rank rigidity conjecture – irreducible locally compact geodesically complete CAT(0) space with geometric action of a group G is either a symmetric space or Euclidean building, or there is some g ∈ G acting as a rank one isometry. There are a lot of partial results concerning those question and I am trying to give (at least partial) generalizations of them. References [1] B. H. Bowditch, A topological characterization of hyperbolic groups. Journal of American mathematical society 11 (1998), No. 3, p. 643–667. [2] P-E. Caprace, N. Monod, Isometry groups of non-positively curved spaces: structure theory. Journal of Topology 2 (2009), No. 4, p. 661–700. [3] P-E. Caprace, G. Zadnik, Regular elements in CAT(0) groups. Preprint on http://arxiv.org/abs/1112.4637. [4] http://www.math.tohoku.ac.jp/∼fujiwara/gs04.2.pdf. [On October 21st 2011.] [5] M. Sageev, D. T. Wise, Periodic ﬂats in CAT(0) cube complexes. Algebraic & Geometric Topology 11 (2011), p. 1793–1820.

CHAPTER 3

(Co-)Homological properties and ﬁnite properties of groups

Francesca Diana Uniformly finite homology was first introduced by J. Block and S. Weinberger in [1] to study the structure of noncompact spaces with bounded geometry; it is a coarse homology theory in the sense that two coarsely quasi-isometric spaces have the same uniformly finite homology. For a uniformly discrete metric space X having bounded uf geometry (UDBG-space) it is the homology of the chain complex (C∗ (X; R), ∂∗) defined as follows:

uf P Cn (X; R) is the vector space of (inﬁnite) sums c = x axx where x = (x0, . . . , xn) ∈ n+1 X (cartesian product of n+1 copies of X) and ax ∈ R that satisﬁes the following properties:

(1) there exists a constant Mc such that |ax| < Mc for every x = (x0, . . . , xn) ∈ Xn+1; (2) there exists a constant Rc such that ax = 0 if d(xi, xj) > Rc for some i 6= j ∈ {0, . . . , n};

The boundary map ∂n is the usual one given by the alternating sum of the “faces” uf (the ith face being (x0,..., xˆi, . . . , xn)). One can also deﬁne Cn (X; Z) in a similar way.

Block and Weinberger give a nice characterization of amenability using uniformly ﬁnite homology; in particular they prove that:

uf Theorem. If X is a UDBG metric space, then H0 (X) = 0 if and only if X is not amenable. Even though some results have been already proved (see, for example, [2]), uniformly ﬁnite homology groups with higher degrees (n > 0) are not yet well understood. In my studies I try to investigate the geometric information contained in the uniformly ﬁnite homology groups in higher degrees and to see if they have nice properties concerning the (non)amenability of metric spaces.

Uniformly ﬁnite homology has also an interesting application in the construction of aperiodic tilings for nonamenable spaces. Block and Weinberger in [1] prove that:

Theorem. If X is a nonamenable, noncompact singular manifold of dimension at least 2, then X has an aperiodic tiling.

13 14 3. (CO-)HOMOLOGICAL PROPERTIES

On the other hand, it is still not so clear if it is possible to construct aperiodic tilings in the amenable case. I am also interested in looking into this problem. References [1] J. Block, S. Weinberger, Aperiodic tilings, positive scalar curvature and amenability of spaces, J. Amer. Math. Soc., 5 (1992), no. 4, 907-918. [2] O.Attie, J. Block, S. Weinberger, Characteristic classes and distortion of diﬀeo- morphisms, J. Amer. Math. Soc. 5 (1992), no. 4, 919-921. [3] K. Whyte, Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture, Duke Math. J. 99 (1999), no. 1, 93-112.

Dieter Degrijse My research interests include algebraic topology, geometric group theory and homological algebra. In my work I enjoy combining ideas and techniques from geometry, such as group actions, CAT(0)-geometry and hyperbolic geometry, with techniques from homological algebra, such as spectral sequences and group cohomology. My PhD research revolves around geometric and cohomological finiteness properties of classifying spaces of groups for families of subgroups. Let G be a discrete group and let F be a family of subgroups of G.A classifying space of G for the family F H or model for EF G, is a G-CW-complex X such that X is contractible for every H in F and is empty when H is not in F. Motivated by the Baum-Connes and Farrell-Jones conjecture, there is a particular interest to study classifying spaces for the families of finite subgroups FIN and virtually cyclic subgroups VC. These conjectures predict isomorphisms between certain equivariant homology theories of classifying spaces of G and K- and L-theories of reduced group C∗-algebras and group rings of G. Other applications of classifying spaces for the family of finite subgroups include computations in group cohomology and a generalization from finite to infinite groups of the Atiyah-Segal completion theorem in topological K- theory. With these applications in mind, it is always desirable to have models for EF G with good geometric properties. One such property is the dimension of EF G. Although a classifying space always exists for any discrete group and family of subgroups, it need not be finite dimensional. The smallest possible dimension of a model for EF G is an invariant of the group called the geometric dimension of G for the family F, and is denoted by gdF G. In [1-3] we have constructed finite dimensional models for EVCG for several classes of groups, including countable elementary amenable groups of finite Hirsch length, generalized Baumslag-Solitar groups, mapping class groups of closed orientable surfaces, and finitely generated linear groups over a field of positive characteristic. In [2] we give a negative answer to a question posed by W. Lück asking whether or not the inequality gdVC(G) ≤ gdFIN (G) + 1 holds for any group G. We do this by constructing a family of integral linear groups of type F for which the geometric dimension for the family of virtually cyclic subgroups is finite but arbitrarily larger than the geometric dimension for the family of finite subgroups. These counterexamples are build by taking products of certain extensions of Bestvina-Brady groups. On the other hand, we show that all countable elementary amenable groups of finite Hirsch length do satisfy the inequality gdVC(G) ≤ gdFIN (G) + 1. JEAN RAIMBAULT 15

References [1] Degrijse, D. and Petrosyan, N., Bredon cohomological dimensions for groups acting on CAT(0)-spaces, submitted (2012) [2] Degrijse, D. and Petrosyan, N., Geometric dimension of groups for the family of virtually cyclic subgroups, submitted (2012) [3] Degrijse, D. and Petrosyan, N., Commensurators and classifying spaces with virtually cyclic stabilizers, Groups, Geometry, and Dynamics (to appear) (2012)

Michal Marcinkowski I am interested in the following two classes of questions: (1) Uniformly finite homology. This is a coarse version of homology theory defined by Block and Weinberger. Its relevance to geometric group theory is justified by the fact that vanishing of the zero homology char- acterises non-amenability. This fact was used to construct an aperiodic systems of tiles for covers of compact manifolds with non-amenable deck transformation groups. It turns out that a simple modification of this method gives rice to tilings of covers whose deck transformation groups are some special amenable groups (such as Grigorchuk groups). Nonetheless, for most amenable groups there are no known homological constructions (or any?) of aperiodic systems of tilings. I am also interested in higher uf homology groups which were recently used to give homological characterisations of geometrical notions such as macroscopic dimension (due to A.Dranishnikov) and topological amenability (die to J.Brodzki, P.W.Nowak, G.A.Niblo, N.Wright). (2) a-T-m property. We say that a group is a-T-m(enable) if it admits a proper affine action on a Hilbert space. Every such an action consists of a linear transformation and a 1-cocycle. I am interested in methods of constructing affine representations as well as in growths of 1-cocycles.

Jean Raimbault My research is mainly concerned with the topology of locally symmetric spaces, and focuses especially on their homology and its behaviour in sequences of such spaces. One motivation is to study the integral homology of finite covers of three– manifolds, especially hyperbolic ones. While the characteristic 0 homology has been well-studied following Thurston’s conjectures, the torsion part of the first homology group has not benefited from such attention until recently. Another nice setting to study these questions is that of congruence manifolds, where additional motivation and insight are provided by number theory. Following are some details about more specific topics.

Growth of torsion homology in abelian covers. One context where the integral homology of finite covering spaces has been the subject of some research is that of “torsion numbers” of knots in the three–sphere, i.e. the orders of the torsion part of the first homology of cyclic covers of their complements. As a further example, if X is a finite CW-complex with an epimorphism φ : π1(X) → Z and Xn, Xb are the associated n-fold and infinite cyclic cover, then for every p the sequence |Hp(Xn)tors| has exponential growth whose rate is computable from a presentation matrix for Hp(Xb). This problem is related to that of approximating 16 3. (CO-)HOMOLOGICAL PROPERTIES combinatorial `2-invariants of infinite abelian covers in the spirit of W. Lück’s seminal paper Approximating L2-invariants by their finite-dimensional analogues.

Integral homology of congruence subgroups of SL2. A special case of a conjecture by N. Bergeron and A. Venkatesh states that if Γ is a lattice in SL2(C) and Γn a sequence of congruence lattices then H1(Γn) should be exponential in the covolume with growth rate 1/6π. When Γ is cocompact they prove a similar result for homology with coefficients in certain nontrivial Γ-modules. The combinatorial tools used to deal with abelian covers are useless in this setting and one has to use their analytic counterparts, most notably Ray-Singer analytic torsion and the Cheeger-MüllerTheorem. An interesting question is to extend this result to a noncompact setting,√ namely to sequences of congruence subgroups in a Bianchi group such as SL2(Z[ −1]). The main difficulty in doing this lies in the fact that the works of Ray–Singer, Cheeger, Müllerand others on analytic and Reidemeister torsions do not deal with noncompact manifolds–in fact the torsions are not even naturally defined for those. Some results can nevertheless be proved. Limit multiplicities and random subgroups. Linked to the last problem is the study of multiplicities of eigenvalues of the Laplacian, or multiplicities of unitary representations, in the spaces of square-integrable forms on finite-volume locally symmetric spaces. For compact ones this is the topic of a joint work with M. Abert, N. Bergeron, I. Biringer, T. Gelander, N. Nikolov and I. Samet where we proved that for sequences of congruence subgroups these multiplicities are sublinear in the volume (except possibly in the middle dimension). One can also prove a similar result for subgroups of Bianchi groups. This topic is also of interest for sequences of noncommensurable manifolds. For example we proved that limit multiplicities hold for any sequence of higher-rank manifolds whose systole is bounded from below. One of the main tools used to prove this last result is the notion of invariant random subgroup (IRS) in a Lie group together with a result of Nevo-Stück-Zimmer on the classification of those in higher-rank groups. We also give various examples of IRSs in rank-one groups, especially in SO(n, 1).

Malte Röer The research begun in my Ph.D.-thesis is focused on questions related to the so-called Gromov-Lawson-Rosenberg conjecture for finite groups and on analytic and geometric aspects of (connective) ko-homology of finite groups. The Gromov- Lawson-Rosenberg conjecture aspires to a classification of compact spin manifolds (with a given group as fundamental group) with a metric of positive scalar curvature. Its formulation involves ko-groups of classifying spaces of the fundamental group. Let π be a finite group. The Atiyah-orientation and the assembly map give a homomorphism Spin αn :Ωn (Bπ) → KOn(Rπ). Geometrically, the homomorphism α can be constructed as the index of a suitable Dirac operator associated with a representative of a spin bordism class. The Gromov-Lawson-Rosenberg conjecture says that a closed spin manifold M of dimension ≥ 5 with π1(M) = π can carry a metric of positive scalar curvature if and only if it represents a class in the kernel of α. The conjecture has so far been proved MARCO SCHWANDT 17 for example in the case where π is trivial (by Stolz) and for groups with periodic cohomology (by Botvinnik, Gilkey and Stolz). In the later case, the ρ-invariant is used to detect classes in the kernel of α. In my thesis, I relate the ρ-invariant to the local cohomology approach developed 1 −∗−1 by Bruner and Greenlees. Let H (KOπ ) be the first local cohomology group of Atiyah-Segal equivariant KO-theory. I define a geometric invariant Spin 1 −n−1 ρˆ :Ωn (Bπ) → H (KOπ ) which turns out to be equivalent to the ρ-invariant used in the proof for groups with periodic cohomology. I use this new construction of ρ to define a more general invariant which can detect certain Bott-torsion classes in kon(Bπ). These classes cannot be found by the classical ρ-invariant which is invariant under multiplication with the Bott-element. In current research projects, I continue to look at aspects of the Gromov-Lawson- Rosenberg conjecture. For example, I am interested in toral classes in the ko- homology of elementary abelian groups and the use of surgery to produce manifolds with positive scalar curvature in these classes. Besides my interest in the Gromov-Lawson-Rosenberg conjecture, I would like to enter new fields like systolic geometry and geometric group theory.

Marco Schwandt My research interests are in geometric group theory and are centered around finiteness properties of groups. The class of groups I am mainly interested in are parabolic subgroups of reductive algebraic groups over function fields. My PhD project, under supervision of Prof. Kai-Uwe Bux, is to determine the finiteness properties of S-arithmetic subgroups of such groups. Another project, joint with Bux, Martin Fluch, Stefan Witzel and Matthew Zaremsky, was concerned with the finiteness properties of generalized Thompson’s groups, such as the braided versions Vbr and Fbr of V and F and the higher-dimensional Brin–Thompson groups sV for s ∈ N. These projects ivolved proving that these groups are of type F∞. Parabolic subgroups: A subgroup P of a Chevalley group G is called parabolic if it contains a Borel subgroup B. Let S denote a finite set of places of the global function field K and OS be the ring of S-integers. By work of Bux, Ralf Khl and Witzel, G(OS) is known to be of type Fd−1 but not of type Fd, where d equals the sum of the local ranks of G over the completions Kp for p ∈ S. On the other hand, Bux proved in his thesis, that the groups B(OS) are of type F|S|−1 but not of type F|S|. In particular the finiteness properties of the Borel subgroups only depend on the number of places, not on the local ranks. It is therefore a natural thing to ask for the finiteness properties of proper parabolic subgroups in between B and G and the behaviour of the finiteness length as the groups P grow. Both of the above results were obtained by considering actions of the groups on affine buildings and using tools like discrete Morse theory to analyze these actions. It stands to reason that the same tools could be used to determine the finiteness properties of the parabolic subgroups. Thompson’s groups: Elements of Thompson’s group F can be described in terms of “strands”. A single strand can split into two and two adjacent strands can merge into one. This point of view leads to the well-known description of F by paired tree diagrams or “split-merge” diagrams. If in addition the strands in these 18 3. (CO-)HOMOLOGICAL PROPERTIES diagrams are allowed to braid with each other, one obtains a description of the braided Thompson’s group Vbr as intoduced by Matt Brin and shown to be finitely presented. In joint work with Bux, Fluch, Witzel and Zaremsky we recently proved that Vbr is even of type F∞. This was done by considering the action of Vbr on a space X that is termed the Stein space for Vbr. This space is a retract of the “natural” Vbr-space, that is the realization of the poset of certain split-braid- merge diagrams. Passing to the Stein space proved to be one of the key steps in proving that Vbr is of type F∞. In particular the descending links in X are closely related to the well-studied matching complexes of graphs and lead to the notion of matching complexes of arcs on surfaces. The same idea of passage to a Stein space also allowed us to prove F∞ for the pure braided Thompson’s group Fbr and also another generalization of V , namely the higher dimensional Brin–Thompson groups sV for s ∈ N. There are further generalizations of Thompson’s groups one can think of. For example a braided version of T , or a twisted version of V by considering “bands” instead of “strands” and allowing them to twist. In theory, our framework should also show these groups to be F∞, once they have been constructed. I would be very interested in knowing if this could indeed be carried out. Additional topics: In connection with past and present projects I am also interested in the theory of buildings, especially affine buildings. Most recently I have −1 become interested in the group SL2(Z[t, t ]) and the question of whether it is finitely generated or not.

Vera Tonić My research so far has been in geometric topology, more precisely it involved res- olution theorems in covering dimension and cohomological dimension theory, but I am also interested in asymptotic dimension theory, which is a large scale analog of covering dimension in coarse geometry ([6]). Asymptotic dimension was introduced by Mikhail Gromov ([5]), and can be defined as follows: for a metric space X and n ∈ N≥0, we say that asdim X ≤ n if every uniformly bounded open cover U of X can be coarsened to a uniformly bounded open cover V of X with multiplicity of V ≤ n + 1 (see [1] for properties). Since asdim is preserved by coarse equivalence between metric spaces, for any finitely generated group Γ its asdim Γ is invariant of the choice of a generating set for Γ ((Γ, dS1 ) and (Γ, dS2 ) being coarsely equivalent, for dS1 and dS2 corresponding word metrics, and S1, S2 finite generating sets for Γ). Asymptotic dimension is useful in investigating discrete groups, and my particular interest is in formulas connecting asdim of a group with the covering dimension dim of its boundary at infinity. For example, in [3], Sergei Buyalo and Nina Lebe- deva have proven the Gromov conjecture that the asymptotic dimension of every hyperbolic group Γ equals the covering dimension of its Gromov boundary plus 1, that is, asdim Γ = dim ∂Γ + 1. However, this equality does not hold for hyperbolic spaces in general (see [4]). The question I would like to investigate is if the analogous formula holds true for a curve complex, that is, if it is true that asdim(C(S)) = dim(∂C(S)) + 1, where C(S) is a curve complex for S = Sg,p = a compact orientable surface of genus g with p punctures. This question is mentioned by Greg Bell and Koji Fujiwara in [2] among open problems, as a possible approach to finding the asymptotic dimension SARAH WAUTERS 19 of a curve graph (and therefore a curve complex). It is known that the asymptotic dimension of a curve complex is finite, since asdim(C(S)) = asdim(C(1)(S)) is finite ([2]).

References [1] G. Bell, A. N. Dranishnikov, Asymptotic Dimension, Topology and its Appls. 155 (2008), 1265–1296. [2] G. Bell, K. Fujiwara, The asymptotic dimension of a curve graph is ﬁnite, arXiv:math/0509216 [3] S. Buyalo, N. Lebedeva, Dimension of locally and asymptotically self-similar spaces, St. Petersburg Math. Jour. 19,(2008), 45–65. arXiv:math/0509433 [4] T. Gentimis, Asymptotic dimension and boundaries of hyperbolic spaces, arXiv:1209.2670 [5] M. Gromov, Asymptotic invariants of inﬁnite groups, Geometric Group Theory, vol. 2, Cambridge University Press, Cambridge (1993) [6] J. Roe, Lectures on coarse geometry, University Lecture Series, vol. 31, AMS, 2003. [7] S. Schleimer, Notes on the complex of curves, available at http://www.warwick.ac.uk/~masgar/math.html.

Sarah Wauters I am interested in the second cohomology of finitely generated, torsion-free nilpotent groups, hereafter called T -groups. In general, the second cohomology of a nilpotent group is an important invariant, since it classifies the extensions and in a way, all nilpotent groups are built up inductively by taking (central) extensions. It is well known that every T -group admits a Mal’cev basis with polynomial group law, i.e. every T -group is isomorphic to Zn as a set, where the group law is polynomial in the entries. It is therefore not surprising that Hartl studied these groups using the polynomial methods (e.g. [1,2]). In his PhD thesis ([1]), Hartl showed that all the 2-cocycles of a T -group with coefficients in a nilpotent module are cohomologous to a polynomial 2-cocycle, as introduced by Passi. Furthermore, he gives an expression of the polynomial cohomology in this case. Let G be a two-step T -group. Denote the lower central series by {γi(G)} and p p p their isolators by { γi(G)}. Write L1 = G/ γ2(G), L2 = γ2(G), and let c : L1∧L1 → L2 be the map induced by the commutator map in G. By investigating certain filtration quotients and using the expression for the second cohomology in [1], we can show the following formula. Theorem (Dekimpe-Hartl-W.). If G is a two-step T -group and M is a trivial G-module that is torsion-free as abelian group, then 2 ∼ H (G, M) = HomZ(Ker(c),M) ⊕ HomZ(L1 ⊗ L2 S, M) ⊕ ExtZ(Cokerc, M), where S is the subgroup of L1 ⊗ L2 generated by all elements x ⊗ c(y ∧ z) + y ⊗ c(z ∧ x) + z ⊗ c(x ∧ y) with x, y, z ∈ L1. In fact, the isomorphism can be made explicit. With a little more effort, one can use the same procedure to give a formula for three-step T -groups (work in progress).

References [1] Hartl, M. Abelsche Modelle Nilpotenter Gruppen. PhD Thesis, RFW-Univ. Bonn, 1991. 20 3. (CO-)HOMOLOGICAL PROPERTIES

[2] Hartl, M. Polynomiality properties of group extensions with a torsion-free abelian kernel, Journal of Algebra vol. 179, 1996, 380 − 415. CHAPTER 4

Analytic, ergodic and probabilistic methods in group theory

Corina Ciobotaru One motivation of Murray and von Neumann to study rings of operators were unitary representations of locally compact groups. Rings of operators are nowadays called von Neumann algebras and group von Neumann algebras are indeed one ma- jor source of examples of von Neumann algebras. However, mostly von Neumann algebras associated to discrete groups were considered by the operator algebras community. This is mostly due to the fact that discrete group von Neumann algebras, though very difficult objects to understand, are still very accessible in terms of the group structure compared to non-discrete groups. We want to find examples of totally disconnected locally compact groups which give rise to interesting group von Neumann algebras. As G.Willis explains in his article form 1994, every locally compact group is an extension of a connected group by a totally disconnected group. Connected locally compact groups can be approximated by connected Lie groups. Indeed, one solution to Hilbert’s fifth problem says that every connected locally compact group is the projective limit of connected Lie groups. A totally disconnected group admits no similar approximation by p-adic Lie groups in general, but it can be build up from easier data in the following sense. If a totally disconnected group admits a compact open and normal subgroup, then it is the extension of a compact group by a discrete group. Even if a totally disconnected group does not admit a normal compact open subgroup, it always possesses a neighbourhood basis of the identity consisting of compact open subgroups. Even more is true in general and this is explained by the concept of tidy subgroups in that article of G. Willis. There are two main sources of examples of totally disconnected locally compact groups. The first class are p-adic Lie groups and more generally algebraic groups. If G is an algebraic group over an algebraically closed, totally disconnected, locally compact field k, then its k-points G(k) inherit a totally disconnected, locally compact topology from k. Specifically, if k = Qp, then we call G(Qp) a p-adic Lie group. Algebraic groups admit two extreme subclasses, the unipotent and the reductive algebraic groups. An example of the first kind is given by the ax+b-group, the typical example of the second kind are general linear groups. Reductive algebraic groups in characteristic 0 satisfy complete reducibility of their linear representations and more generally Haboush’s theorem describes geometric reductivity of linear representations of reductive algebraic groups in characteristic p. Another source of totally disconnected groups are automorphism groups of discrete structures. It is known that a Polish group has a neighbourhood basis of open subgroups if and only if it is the automorphism group of a countable structure. (Such Polish groups are

21 22 4. ANALYTIC, ERGODIC & PROBABILISTIC METHODS called non-Archimedean.) So it is natural to consider such automorphism groups as a source of totally disconnected, locally compact groups. The best known example in this class is Aut(Td), the automorphism group of a d-regular tree. But let us point out that a big class of totally disconnected locally compact groups does not give rise to new examples of von Neumann algebras. Theorem. The group von Neumann algebra of every reductive algebraic group over a non-Archimedian local ﬁeld is of type I. Note that von Neumann algebras of type I are injective so in this context the main objective of this joint project is to study new classes of totally disconnected locally compact groups from the point of view of their von Neumann algebras and so to come up with examples of non-injective group von Neumann algebras of totally disconnected locally compact groups.

Jonas Deré One of the first examples of hyperbolic dynamical systems is given by the so-called Arnold’s cat map. This map is the diffeomorphism of the torus T 2 which is induced by the matrix 2 1 A = . 1 1 Note that A only has eigenvalues of absolute value different from 1. This map has many interesting properties, e.g. one can check that the periodic points are dense in T 2 and that it’s structurally stable. Recall that a diffeomorphism f is called structurally stable if all of its small perturbations g are topologically conjugate to f, i.e. if there exists a homeomorphism h such that f = h−1gh for all those g. This example was generalized to a much larger class of diffeomorphisms by D. Anosov, who proved that all diffeomorphisms in this class are structurally stable (see [1]). In honor of Anosov’s work, these maps were called Anosov diffeomorphisms and they are intensively studied because of their interesting dynamical properties.

Definition. An Anosov diffeomorphism f : M → M on a closed manifold M is a diffeomorphism that satisfies the following properties: (1) There is a continuous splitting of the tangent bundle TM = Es ⊕Eu which is invariant under the derivative df. (2) There exist a real constant λ > 1 and a Riemannian metric k · k such that kdf n(v)k ≤ λ−nkvk if v ∈ Es kdf n(v)k ≥ λnkvk if v ∈ Eu. This means that f exponentially contracts all tangent vectors of Es and exponentially expands all elements of Eu. It is easy to see that Arnold’s cat map is Anosov by looking at the eigenspaces of A.

Just like Arnold’s cat map, all other known examples of Anosov diffeomorphisms are of algebraic nature. For example, every Anosov diffeomorphism on a n-dimensional torus is topologically conjugate to a diffeomorphism induced by an integer matrix on the universal cover Rn [3]. It is conjectured that every Anosov diffeomorphism is topologically conjugate to a hyperbolic infra-nilautomorphism, where RIIKKA KANGASLAMPI 23 infra-nilmanifolds form the natural generalization of tori.

My research is focused on the classification of all compact manifolds (up to homeomorphism) which admit an Anosov diffeomorphism. Since all manifolds with Anosov diffeomorphisms are conjectured to be homeomorphic to an infra-nilmanifold, we focus our attention on these manifolds. Most of the results up to now are about nilmanifolds, which can be studied by looking at rational Lie algebras and their automorphisms [2]. But there are only very few results about when an infra- nilmanifold admits an Anosov diffeomorphism, except for the case of flat manifolds, where a complete classification was given by Porteus in [4]. Currently I’m working on the classification of all infra-nilmanifolds admitting an Anosov diffeomorphism which are of dimension 6 or which are modeled on a free nilpotent Lie group.

References [1] Anosov, D.V., Geodesic flow on closed Riemannian manifolds with negative curvature, Proc. Steklov Inst. (1969) [2] Lauret, J. & Will, C. E., On Anosov automorphisms of nilmanifolds, Journal of Pure and Applied Algebra (2008) [3] Manning, A., There are no new Anosov diffeomorphisms on tori, American Journal of Mathematics (1974) [4] Porteus, H. L., Anosov diffeomorphisms of flat manifolds, Topology (1972)

Riikka Kangaslampi My research activities can be divided into two groups: study of uniformly quasiregular mappings, mainly their existence and dynamics on manifolds, and study of groups acting on hyperbolic buildings. In these two areas the studied objects are quite diﬀerent and the dynamics mutually exclusive, but similar, somewhat combinatorial research methods are used in both projects.

Uniformly quasiregular mappings. A uniformly quasiregular mapping (uqr mapping) is a mapping f : M → M acting on a compact Riemannian manifold M in such a way that all the iterates are quasiregular with a uniform bound on the distortion constant. In my doctoral thesis I constructed uqr mappings with this analogue of Lattès’construction on all elliptic three-dimensional Riemannian manifolds. Later with L. Astola and K. Peltonen we studied uniformly quasiregular mappings of Lattèstype more generally on compact Riemannian manifolds. We showed that there is a plenitude of compact manifolds in dimensions 4, 5 and higher that support these mappings. Recent result with Peltonen and J. -M. Wu presents first example of a uqr mapping with 2-torus as the Julia set. In the future I plan to study the existence properties of uniformly quasiregular mappings of Lattèstype further. The goals on longer perspective are to classify manifolds that support Lattès-type uqr mappings and to categorize the possible Julia sets that these mappings can have.

Hyperbolic buildings. My research on the area of hyperbolic buildings is joint work with Dr. Alina Vdovina from Newcastle University. Our research concentrates on hyperbolic groups and their applications. Together with L. Carbone and Vdovina we constructed and classiﬁed groups acting simply transitively on vertices of hyperbolic triangular buildings of the smallest non-trivial thickness, having 24 4. ANALYTIC, ERGODIC & PROBABILISTIC METHODS the minimal generalized quadrangle as a link. We obtained both torsion and torsion free groups acting on the same building. We gave also a method to obtain groups acting cocompactly on hyperbolic buildings with n-sided chambers for arbitrary n ≥ 3, starting from one of their groups on a triangular building. The aim of the research on hyperbolic buildings is to construct new buildings and groups acting on buildings using so called polygonal presentations, and to study the structure of buildings with the help of polygonal presentations. Namely, to study periodic apartments in hyperbolic buildings, and to apply this to construct new dynamical systems associated to hyperbolic buildings.

Juhani Koivisto My aim is to do postgraduate studies and research leading to a doctoral degree at the Department of Mathematics and Statistics at University of Helsinki under the supervision of dos. Ilkka Holopainen in the research group of ”Analysis, metric geometry, and differential and metric topology”. My area of research is curvature in metric spaces, coarse geometry, and rigidity properties of groups. I have worked on the so called asymptotic Dirichlet problem, in particular trying to find some curvature or other condition for metrizability of the boundary of CAT(0) spaces. I also work on generalizing property (T ) to reflexive Banach spaces, which so far has produced the following application [1]: Theorem. Let X be a locally finite 2-dimensional simplicial complex, Γ a discrete properly discontinuous group of automorphisms of X and π :Γ → B(H) a uniformly bounded representation of Γ on a separable infinite-dimensional Hilbert space H. If for any vertex τ of X the link Xτ is connected and √ 2 sup kπgk < , g∈Γ κ2(Xτ , H) then L2H1(X, π) = 0. Motivating this, among others, is Shalom’s conjecture [2] stating that any hyperbolic group Γ admits a uniformly bounded representation π with H1(Γ, π) 6= 0 together with a proper cocycle in Z1(Γ, π).

References [1] J. Koivisto, Automorphism groups of simplicial complexes and rigidity for uniformly bounded representations, preprint arXiv:1208.6476v2 [math.GR]. [2] Oberwolfach Report, no. 29/2001.

Nicolas Matte Bon I am currently starting a Ph.D. about random walks on groups and their interactions with geometric group theory. Most likely, I will start by focusing around behavior of the rate of escape of random walks on discrete amenable groups. Given a symmetric measure µ on a group G equipped with a word metric, let gn be the corresponding random walk. One can look to the asymptotic behavior of the mean distance E(|gn|), or more generally of the whole random variable |gn|. On non-amenable groups it is well known that they both grow linearly, because of non-triviality of the Poisson boundary. On amenable groups however the measure µ might have trivial boundary, and a huge ADRIANA NICOLAE 25 variety of sub-linear behaviors are possible (it is still an open question to describe all the possible behaviors, but some important partial answers are known). It is then interesting to try to relate this behavior to algebraic and geometric properties of the group, and to understand to what extent it is possible to obtain general results for large classes of groups. For instance, on groups of polynomial growth it 1 is known that for any simple random walk the mean distance grows like n 2 . An example of a natural question: which is the optimal, reasonably defined class of groups with this property? This topic has been increasingly studied in the last decade. I will probably also be interested on questions concerning triviality/non triviality of the boundary for random walks, and the identification of the boundary in different contexts.

Adriana Nicolae My research focuses on analyzing regularity properties of geodesic spaces with the goal to extend the theory of less regular frameworks than those normally considered in nonlinear analysis and ergodic theory and to apply the findings to specific problems in these fields. During my PhD studies I mainly centered on the class of metric spaces where a convexity structure is defined and in which every descending sequence of nonempty, bounded, closed and convex subsets has a nonempty intersection. Such spaces include reflexive Banach spaces, complete CAT(0) spaces, hyperconvex spaces or complete uniformly convex metric spaces with a specific modulus of uniform convexity. Among the addressed research directions are the study of • geometrical properties of these spaces in order to draw conclusions about the existence and uniqueness of fixed points for different classes of mappings [R. Espinola, P. Lorenzo, A. Nicolae, Fixed points, selections and common fixed points of nonexpansive-type mappings, J. Math. Anal. Appl., 382 (2011), 503-515] • the minimal and maximal distance between pairs of sets in geodesic spaces under different conditions for the sets. In [R. Espinola, A. Nicolae, Mu- tually nearest and farthest points of sets and the Drop Theorem in geodesic spaces, Monatsh. Math., 165 (2012), 173-197] we studied the well- posedness of the minimization and maximization problem between two sets in geodesic spaces under different conditions for the sets • regularity properties of geodesic Ptolemy spaces with applications to metric fixed point theory [R. Espinola, A. Nicolae, Geodesic Ptolemy spaces and fixed points, Nonlinear Anal., 74 (2011), 27-34; R. Espinola, A. Nico- lae, Uniform convexity of geodesic Ptolemy spaces, J. Convex Anal. (in press)]. One of the problems I am currently interested in is the study of barycenter techniques in non-positively and non-negatively [S. Ohta, Barycenters in Alexandrov spaces of curvature bounded below, Adv. Geom. (in press)] curved spaces and their different uses in generalizing results from classical ergodic theory. Barycen- ters were used in [T. Austin, A CAT(0)-valued pointwise ergodic theorem, J. Topol. Anal, 3 (2011), 145-152] to prove an extension of the pointwise ergodic theorem for functions with values in a separable complete CAT(0) space, while in [A. Navas, 26 4. ANALYTIC, ERGODIC & PROBABILISTIC METHODS

An L1 ergodic theorem with values in a non-positively curved space via a canonical barycenter map, Ergodic Theory Dynam. Systems (in press)] the considered setting is the one of Busemann convex spaces. At the same time I am interested in the multiplicative ergodic theorem obtained by [A. Karlsson, G. Margulis, A multiplicative ergodic theorem and nonpositively curved spaces, Comm. Math. Phys., 208 (1999), 107-123] with the aim of proving it in a more general context.

Tomasz Odrzygó´zd´z I am a masters student at University of Warsaw, currently I am working on my masters thesis under supervision of Dr. Piotr Przytycki. My thesis will cover some topics in Random Group Theory. Gromov introduced in 1993 the notion of a random finitely presented group on m ≥ 2 generators at density d ∈ (0, 1). The idea was to fix a set of m generators and consider presentations with (2m − 1)dl relations, each of which is a random reduced word of length l. In this model it is very reasonable to investigate properties of random groups when l goes to infinity. Our modification of Gromov’s idea is a square model: length of relations in a presentation is always 4, but we let the number of generators go to infinity. We say that a square model has density equal to d iff the number of drawn relations equals the number of all possible relations to the power d. We have proved two interesting results: 1 • when the density is greater than 2 , the probability that a random group in the square model is trivial tends to 1 when the number of generators converges to infinity. 1 • when the density is less than 4 , the probability that a random group in the square model is free tends to 1 when the number of generators converges to infinity. 1 1 Now we are going to investigate what happens when density is between 4 and 2 . We would like to answer questions like: what is the probability that a random group has Kazhdan’s property (T) or is it possible to cubulate random group?

Alessandro Sisto See page 33.

Werner Thumann In 1986 Novikov and Shubin introduced a number for Riemannian manifolds M with a cocompact proper free action of a group Γ by isometries. Namely, they observed the asymptotic behaviour of the expression Z −t∆p θp(t) := trC e (x, x) dvolx F

−t∆p for t → ∞. Here ∆p is the Laplacian acting on complex p-forms, e ( , ) is the integral kernel for the operator e−t∆p and F is a fundamental domain for the Γ-action. For example, the p-th Novikov-Shubin number is n if θp(t) attains its limit asymptotically as t−n for t → ∞. So these numbers measure how fast heat decays on such manifolds. Later Gromov showed that this number is in fact homotopy invariant, a remarkable result in view of this definition. In 1999 Lück, Reich and Schick introduced an algebraic definition for this invariant which applies STEFFEN WEIL 27 to arbitrary Γ-spaces and also to groups. If G is a (discrete, countable) group, then there is the associated von Neumann algebra N (G) and the L2-homology

(2) ZG Hn (G) := Hn G, N (G) := Torn Z, N (G) Furthermore, there is a notion called capacity for modules over N (G) which measures the size of modules M with M ⊗ U(G) = 0 faithfully (U(G) is the ring of affiliated operators). This class of modules is the torsion class of the torsion theory with flat modules as torsion-free modules. See [1] for more details. Now the Novikov-Shubin invariants NSp(G) of G can be defined as the capacity of the mod- (2) ules Hn (G). As the original definition of Novikov and Shubin suggests, there are connections between these invariants and the large scale geometry of a group G. For example, the 0’th Novikov-Shubin invariant of finitely generated groups is invariant under quasi-isometry due to the following result for an infinite group G  growth(G) ∈ G virtually nilpotent  N NS0(G) = ∞ G amenable but not virtually nilpotent ∞+ G not amenable

In [2] Sauer showed (together with a result of Lia Vaˇs):If G1 and G2 are finitely generated, amenable and quasi-isometric, then their Novikov-Shubin invariants agree in every degree. In unpublished notes of 2010 Shin-Ichi Oguni extended this result to coarsely equivalent groups and showed the coarse-invariance of the vanishing of the Novikov-Shubin numbers for (not necessarily amenable) groups. In my studies I try to find new geometric interpretations for higher degree Novikov- Shubin invariants and investigate further the relationship between these invariants and the notion of quasi-isometry. References [1] W. Lück, H. Reich, T. Schick, Novikov-Shubin invariants for arbitrary group actions and their positivity. Tel Aviv Topology Conference: Rothenberg Festschrift (1998), pages 159-176, Amer. Math. Soc., Providence, RI, 1999. [2] R. Sauer, Homological invariants and quasi-isometry. Geom. Funct. Anal., Volume 16, Number 2, 476-515, 2006.

Steffen Weil My research interest lies in questions which combine dynamics and geometry. Re- cently I have been studying Schmidt games. Winning sets of Schmidt’s game enjoy a remarkable rigidity. Therefore, this game (and modifications of it) have been applied to many examples of complete metric spaces (X, d) to show that the set of ’badly approximable points’, with respect to a given collection of resonant sets in X, is a winning set. For these examples, strategies were deduced that are, in most cases, strongly adapted to the specific dynamics and properties of the underlying setting. In [4], I introduced a new modification of Schmidt’s game which is a combination of the ones of [1] and [2]. This modification allows to state conditions on the collection of resonant sets under which there always exists a winning strategy. Moreover, I discussed properties of winning sets of this modification. For instance, let Γ be a discrete subgroup of the isometry group of a proper CAT(- 1) space and Γ0 be a subgroup of Γ, say Γ0 is bounded parabolic. Concerning the 28 4. ANALYTIC, ERGODIC & PROBABILISTIC METHODS above conditions on the resonant sets, one needs to study the distribution of the limit set of Γ0 under the action of Γ. As a prominent example, let Γ ⊂ I(Hn) be a lattice in the isometry group of the n n ¯ n−1 real hyperbolic half-space model H with one cusp at ∞ ∈ ∂∞H = R . In this case, a sufficiently small horoball H based at ∞ is (Γ-) precisely invariant and we obtain a disjoint collection of horoballs γ(H), γ ∈ Γ − StabΓ(∞), based at γ(∞). The distribution of the orbit points {γ(∞): γ ∈ Γ} ⊂ R¯ n−1 is therefore ”nicely distributed” with respect to the Euclidean radii of these horoballs. The set of points x ∈ Rn for which the geodesic, starting a fixed point in Hn and ending at x, has bounded penetration lengths in the collection of the horoballs turns out to be a winning set in Schmidt’s game. In particular, it is of full Hausdorff-dimension, although of Lebesgue measure zero. Assuming instead that ∞ is the endpoint of an axis α of a hyperbolic isometry in Γ, one can show the similar result, using again that the orbit points γ(∞), γ ∈ Γ, are nicely distributed. That is, the set of points x ∈ Rn for which the geodesic, starting a fixed point in Hn and ending at x, has bounded penetration lengths in the -neighborhood of the images of γ(α), γ ∈ Γ, turns out to be a winning set in Schmidt’s game; again of full Hausdorff-dimension but of Lebesgue measure zero. Denote this set of points by Sα. As a further consequence of winning sets, the intersection ∩Sα, α an axis of a hyperbolic isometry in Γ, is of full Hausdorff- dimension. If Γ is cocompact, then in [3], I proved the existence of special geodesics with endpoints in ∩Sα for which the ”approximation constants” can be estimated by lower bounds. References [1] D. Kleinbock and B. Weiss. Modified schmidt games and diophantine approximation with weights. Israel J. Math., 149, 2005. [2] C. T. McMullen. Winningsets, quasiconformal maps and diophantine approximation. GAFA Goemetric and Functional Analysis, 20, 2010. [3] V. Schroeder and S. Weil. Aperiodic sequences and aperiodic geodesics. Arxiv, 2012. [4] S. Weil. Schmidt Games and Conditions on Resonant Sets, Arxiv, 2012 CHAPTER 5

Systolic and (relatively) Gromov-hyperbolic groups

Dominik Gruber My research is focused on lacunary hyperbolic groups arising from infinite graphical small cancellation presentations. Small cancellation theory is a combinatorial and geometric tool that has been used in the past to construct groups with extreme properties, often counterexamples to conjectures. Many of these groups are certain infinitely presented limits of hyperbolic groups known as lacunary hyperbolic groups. Small cancellation theory for labelled graphs was introduced by Gromov, generalizing existing notions of small cancellation theory. Gromov applied a geometric version of this new theory to construct Gromov’s monster, a lacunary hyperbolic group that coarsely contains an expander graph [1]. This group is currently the only known group that does not coarsely embed into a Hilbert space. It is a counterexample to the Baum-Connes conjecture with coefficients [3] and thus a natural potential counterexample to the Baum-Connes conjecture. I am interested in the combinatorial interpretation of graphical small cancellation theory. In this context, the non-metric C(6) and C(7) small cancellation conditions are natural. I will briefly sketch them: Let S be a finite set. A labelled graph Γ over the alphabet S is a graph each edge of which is assigned an orientation and an element of S. The group defined by Γ, denoted G(Γ), is the group given by the presentation

G(Γ) := hS | words read on closed paths in Γi. A piece on Γ is a labelled path that can be read from two distinct vertices. Let n ∈ N. The labelled graph Γ satisfies the C(n) small cancellation condition if no closed path on Γ is the concatenation of fewer than n pieces. In a recent paper, I established basic properties of groups with (finite and infinite) graphical C(6) and C(7) presentations [2]. Besides generalizations of fundamental facts about classical C(6) and C(7) small cancellation groups, the following are main results:

Theorem Let Γ be a C(7)-labelled graph. Then G(Γ) contains a rank 2 free subgroup unless it is trivial or inﬁnite cyclic.

Theorem Let (Γn)n∈N be a sequence of ﬁnite, connected labelled graphs such that their disjoint union Γ is C(6)-labelled. Then the coarse union of the Γn coarsely and injectively embeds into Cay(G(Γ),S). If, moreover, Γ is C(7)-labelled, there exists an inﬁnite subsequence of graphs (Γkn )n∈N such that G(tn∈NΓkn ) is lacunary

29 30 5. SYSTOLIC AND (RELATIVELY) GROMOV-HYPERBOLIC GROUPS hyperbolic.

Thus, the C(7) condition permits the construction of non-amenable, lacunary hyperbolic groups that coarsely contain prescribed infinite sequences of finite graphs. Most existing constructions of graphical small cancellation groups with extreme properties use geometric small cancellation conditions and rely on probabilistic arguments, whence they are non-explicit. I am currently trying to make explicit constructions of such groups using the more easily accessible non-metric small cancellation conditions. To this end, I have been studying graph-theoretic notions; expander graphs in particular have drawn my interest. My research is supervised by my advisor Goulnara Arzhantseva and supported by her ERC grant “ANALYTIC” no. 259527. References [1] M. Gromov, Random walk in random groups, Geom. Funct. Anal. 13 (2003), no. 1, 73–146 [2] D. Gruber, Groups with graphical C(6) and C(7) small cancellation presentations, submitted, arXiv:1210.0178 (2012). [3] N. Higson, V. Lafforgue and G. Skandalis, Counterexamples to the Baum-Connes conjecture, Geom. Funct. Anal. 12 (2002), no. 2, 330–354.

David Hume The main focus of my work currently is the metric geometry of finitely generated groups G = hSi. One productive method of studying the metric geometry of groups is to obtain “good” embeddings into “nice” metric spaces, for instance coarse embeddings into `p spaces or quasi-isometric embeddings into finite products of trees. Having such embeddings yields strong consequences in other areas of mathematics, most notably in topology and K-theory. 0 A map φ :(G, dS) → (Y, d ) is called a coarse (or uniform) embedding if there exist functions ρ± : R≥0 → R≥0 such that ρ±(n) → ∞ as n → ∞ and ρ−(d(x, y)) ≤ 0 d (φ(x), φ(y)) ≤ ρ+(d(x, y)) for all x, y ∈ X. Any finitely generated group which admits a coarse embedding into some `p space with p ∈ (1, ∞) satisfies the Novikov and coarse Baum-Connes conjectures. p ∗ A stronger notion is the ` compression exponent, αp(X), defined as the supremum over α ∈ [0, 1] such that there exists a coarse embedding φ : G → `p(N) such that −1 α ρ+(n) ≤ Cn + C and ρ−(n) ≥ C n − C. The `p compression exponent of a finitely generated group is closely linked to amenability, the speed of random walks on the Cayley graph and notions of large- scale dimension. Many groups are known to admit coarse embeddings into `p spaces, for example amenable groups, RAAGs and relatively hyperbolic groups whose parabolic subgroups admit such embeddings. Even when it is known that groups do coarsely embed it is in general difficult to calculate their compression exponent and harder still to exhibit an explicit embedding displaying this information. `p compression exponents are known in general for hyperbolic and polycyclic groups, as well as for certain wreath products. The recent paper [1] represents a breakthrough in understanding the metric geometry of mapping class groups G = MCG(Sg,n). They prove that there exists a finite ALEXANDRE MARTIN 31 collection of hyperbolic metric spaces C(Y) built from curve complexes C(Y ) each of which G acts on, such that orbits of the combined action on the product yields an equivariant quasi-isometric embedding G → Q C(Y). The versatility of this construction is shown in [4], where the authors prove that relatively hyperbolic groups embed into the product of the coned-off graph with a quasi-tree of spaces C(H) where the C(H) are cosets of maximal parabolic subgroups. Such spaces are clearly not hyperbolic in general, so the general metric geometry of such spaces C(Y) then becomes an important hurdle in understanding the geometry of mapping class groups and relatively hyperbolic groups. Theorem. [3] Every space C(Y) satisfying the axiomatisation in [1] is quasi- isometric to a tree-graded space T (Y) with pieces T (Y ) uniformly quasi-isometric to C(Y ).

We recall that a geodesic metric space X is tree-graded with pieces {Xi |i ∈ I } if and only if

• for all i 6= j, |Xi ∩ Xj| ≤ 1 and • every simple geodesic triangle (a simple loop consisting of three geodesic edges) is contained in a single piece. To prove this theorem, we introduce a relative version of Manning’s bottleneck property (the relative bottleneck property) and prove that for geodesic metric spaces this is equivalent to being quasi-isometric to a tree-graded space. The consequences for the metric geometry of mapping class groups are the following: Corollary. [3] Mapping class groups of closed surfaces quasi-isometrically embed into ﬁnite products of simplicial trees. Also, they admit explicit embeddings into `p spaces (for each p ≥ 1) exhibiting `p compression exponent 1 and have ﬁnite Assouad-Nagata dimension. We also obtain the following new results for relatively hyperbolic groups.

Corollary. [2, 3] Let G be hyperbolic relative to {H1,H2,...,Hn}. Then, for all p ≥ 1, there is an explicit embedding of G into some `p space exhibiting ∗ ∗ the optimal compression exponent αp(G) = mini{αp(Hi)}. Moreover, G has finite Assouad-Nagata dimension if and only if each Hi does. References [1] Mladen Bestvina, Ken Bromberg, and Koji Fujiwara, The asymptotic dimension of mapping class groups is finite, Available from arXiv:1006.1939v2. [2] David Hume, Direct embeddings of relatively hyperbolic groups with optimal `p compression exponent, Preprint (arXiv:1111.6013v1), 2011. [3] David Hume, Embedding mapping class groups into finite products of trees., Available from arXiv:1207.2132v2, 2012. [4] John M. Mackay and Alessandro Sisto, Embedding relatively hyperbolic groups in products of trees..Available from arXiv:1207.3008v1, 2012.

Alexandre Martin A very general problem in geometric group theory which is at the heart of my research is the following: Given a group G acting cocompactly by simplicial isometries on a simply-connected simplicial complex X such that every stabiliser satisﬁes a given property (P), are there conditions on the geometry of X and on the action 32 5. SYSTOLIC AND (RELATIVELY) GROMOV-HYPERBOLIC GROUPS that ensure that G also satisﬁes property (P)? I have primarily been dealing with actions on CAT(0) simplicial complexes such that every simplex stabiliser admits a boundary in the sense of Bestvina, a structure that appears to have deep impli- cations in geometric topology and manifold theory. I was able to construct such a boundary for G under various conditions of geometric, dynamical and algebraic natures. By studying the dynamics of the group on its boundary, I proved a combination theorem for hyperbolic groups in the case of an acylindrical simple complex of groups of arbitrary dimension :

Combination Theorem for Hyperbolic groups Let G(Y) be a strictly devel- opable simple complex of groups over a ﬁnite simplicial piecewise Euclidean complex. Assume that: • The universal covering of G(Y) is hyperbolic and CAT(0), • The local groups are hyperbolic, and all the inclusions are quasiconvex embeddings, • The action of π1(G(Y)) on the universal cover of G(Y) is acylindrical.

Then π1(G(Y)) is hyperbolic. Furthermore, the local groups embed in π1(G(Y)) as quasiconvex subgroups.

An example of action on a 2-dimensional complex comes from metric small cancellation groups over a graph of groups, for which I proved the following:

Theorem Let G(Γ) be a finite graph of groups over a finite simplicial graph Γ, with fundamental group G. Let R be a finite symmetrized collection of hyperbolic 00 1 elements of G satisfying the small cancellation condition C ( 6 ) for the action of G on the Bass-Serre tree of G(Γ). Assume that: • The local groups are hyperbolic, and all the inclusions are quasiconvex embeddings, • The action of G on the associated Bass-Serre tree is acylindrical. Then G/hhRii is hyperbolic and the local groups of G(Γ) embed as quasiconvex subgroups.

Alessandro Sisto Relatively hyperbolic spaces. Motivating examples of relatively hyperbolic groups include fundamental groups of finite volume hyperbolic or negatively curved manifolds, and hence many fundamental groups of knot complements, fundamental groups of non-geometric 3-manifolds with at least one atoroidal JSJ piece, limit groups, etc. I characterized relative hyperbolicity in terms of projections on peripheral sets and gave a distance formula for relatively hyperbolic groups similar to the one for mapping class groups. I also extended several definitions of relative hyperbolic from the context of groups to the context of metric spaces and showed a relative thin triangle condition in relatively hyperbolic spaces. The main tool is a lemma that allows to recognize relative hyperbolicity in a space where a family of paths has been assigned. Applications of the techniques include a new, simpler characterization of PETRA SCHWER 33 hyperbolically embedded subgroups as defined by Dahmani, Guiradel and Osin and results on the divergence function of relatively hyperbolic groups. In a joint work with John MacKay, we showed several geometric properties of boundaries of relatively hyperbolic group and in particular constructed quasi-symmetric embeddings of circle in certain such boundaries. Our result, not in its most general form, is that a relatively hyperbolic group contains a quasi-isometrically embedded copy of the hyperbolic plane if and only if G does not split as a non-trivial graph of groups where the edge groups are finite and the vertex groups virtually nilpotent.

Random walks and group actions. I deﬁned a notion of (weakly) contracting element of a group. Examples include hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, iwips in Out(Fn), rank one isometries in groups acting properly on proper CAT (0) spaces, elements acting hyperbolically on the Bass-Serre tree in fundamental groups of acylindrical graphs of groups. This notion is therefore very general and is very much related to the notion of hyperbolically embedded subgroup introduced by Dahmani, Guirardel and Osin. I showed that if the group G is not virtually cyclic and contains a contracting element, then the probability that a simple random walk supported on G gives rise to a non-contracting element decays exponentially in the length of the random walk. Informally speaking, the result is that contracting elements are generic.

Current work. I am working on a few projects related to boundaries of relatively hyperbolic groups, including understanding boundaries of mapping tori and Dehn ﬁllings, comparing boundaries of diﬀerent peripheral structures of the same relatively hyperbolic group and characterizing boundary extensions of quasi- isometric embeddings. On the random walks side, I am working on further results on random walks for groups with contracting elements acting on, say, a hyperbolic space, including showing positive drift and convergence to a boundary point.

Petra Schwer Admitting a geometric action on a CAT(0) space has many algebraic consequences for a group. Such groups have solvable word or conjugacy problem, a quadratic Dehn function and are biautomatic. Hence on the one hand given a group one does very much dare to construct a CAT(0) space on which the group acts geometrically. Also one wishes to understand the general structure of non-positively curved spaces. Gromov’s surprisingly simple characterization of locally CAT(0) cubical complexes (their links have to be ﬂag simplicial complexes) lead to the question whether there is a comparable characterization of non-positive curvature for metric simplicial or other polyhedral complexes. However it turns out that curvature testing for simplicial complexes is hard and only in small dimensions explicit algorithms are known (compare the results by Elder–Mc Cammond). Januszkiewicz–Swi¸atkowski´ and Haglund developed the notion of simplicial nonpositive curvature (SNPC) and systolic spaces describing curvature in an easy to check manner via diameters of links. Although SNPC is (in general) neither im- plied by nor does it imply the CAT(0) property, these systolic spaces share many properties with CAT(0) spaces. 34 5. SYSTOLIC AND (RELATIVELY) GROMOV-HYPERBOLIC GROUPS

A long term goal would be to extend the known results or to develop alternative methods for curvature testing in nice classes of metric polyhedral complexes. Fur- ther I would like to understand which classes of groups are systolic. In a joint project with Piotr Przytycki we were able to show, that triangle groups (as well as certain 3-dim. hyperbolic groups) can be systolized. One of the main classes of CAT(0) spaces are affine buildings which were originally introduced by Jaques Tits to assign geometric objects to algebraic groups over valued fields. Previously I have been working on affine buildings and their non- discrete generalizations with the goal to understand their geometric behavior and general axiomatic structure. Looking like a useless toy the Λ-buildings (where the defining valued field is “replaced by” an arbitrary ordered abelian group with Krull valuation) turn out to be quite a useful object. Not only are they functorial in Λ but together with Struyve we were able to apply them to obtain a new proof of the fact that asymptotic cones of R-buildings are again such. A fact which was previously shown by Kleiner–Leeb with completely different methods.

Krzysztof Swiecicki´ I am a second-year master’s student at the University of Warsaw. My first contact with GGT was during the first Young Geometric Group Theory Meeting at Bedlewo and I have been interested in it ever since. Currently I am working under the supervision of Pawel Zawi´slakon my master thesis, which concerns proving the analogue of Helly’s Theorem for systolic complexes. For geodesic metric space X, we define the Helly number h(X) to be the smallest natural number such that any finite family of h(X)-wise non-disjoint convex subsets of X has a non-empty intersection. The Helly dimension of X is equal to the Helly number minus one. Classical Helly’s Theorem for Euclidean spaces states that the Helly dimension of the Euclidean space Rd equals its dimension as a real vector space. There is also a well known result for CAT (0) cube complexes, which states that, regardless their topological dimension, they all have the Helly dimension equal to one (we demand convex subsets in our definition to be convex subcomplexes). The notion of k-systolic complexes (k ≥ 6 is a natural number) was introduced by Tadeusz Januszkiewicz and Jacek Swiatkowski´ in [4] and independently by Frederic Haglund in [3]. Systolic complexes (systolic means 6-systolic) are simplicial analogues of nonpositively curved spaces and inherit lots of CAT (0)-like properties. For example: they are contractible ([4]), the analogue of Flat Torus Theorem holds for them ([2]) and every finite group acting geometrically on a systolic complex by simplicial automorphisms has global fix point ([1]). One would expect for systolic complexes a similar kind of Helly’s like properties as for CAT (0) cube complexes. However, let us consider an example of a single n-dimensional simplex with its codimension 1 faces being a family of convex subcomplexes. Then the intersection of any subfamily of the cardinality n is non-empty, but the intersection of the entire family is empty. We expect that it is the worst case to happen, namely in systolic complexes instead of nonempty intersection of a family of convex subcomplexes we would rather expect the existence of a single simplex with a nonempty intersection with all these subcomplexes. So far we were able to prove that a 7-systolic complex has the Helly dimension equal to one. Namely, for any three convex subcomplexes X1,X2,X3 which are pairwise intersecting, there exists a simplex in X having a nontrivial intersection with all of KRZYSZTOF SWIECICKI´ 35 them. We expect that all systolic complexes have Helly dimension equal to two.

References [1] V. Chepoi, D. Osajda, Dismantlability of weakly systolic complexes and applications, submitted. [2] T. Elsner, Flats and the ﬂat torus theorem for systolic spaces, Geometry & Topology 13 (2009), 661-698. [3] F. Haglund, Complexes simpliciaux hyperboliques de grande dimension, preprint, Prepublication Orsay 71 (2003). [4] T. Januszkiewicz, J. Swiatkowski,´ Simplicial nonpositive curvature, Publ. Math. IHES, 104 (2006), 1-85.

CHAPTER 6

Surface groups, mapping class groups and Out(Fn)

Valentina Disarlo My research interests are in the fields of geometric topology and low-dimensional topology, especially Riemann surfaces, mapping class groups, moduli spaces and the geometry of the Teichmüllerspace. I am particularly interested in the interplay between Teichmüllertheory and geometric group theory and in the combinatorial models used to study the large scale features of the Teichmüllerspace with its different metrics.

Matthew Durham I am a fourth year PhD student working under Daniel Groves. My research is broadly in geometric group theory, more specifically in Teichmuller theory and mapping class groups of surfaces. In particular, I have been studying the structure of fixed and almost-fixed point sets of the action of finite subgroups of the mapping class group on Teichmuller space in the Teichmuller metric (such sets exist by Kerckhoff’s solution to the Nielsen realization problem). My methods are coarse and involve the hierarchy machinery of Masur-Minsky and Rafi’s adaptation of that machinery to Teichmuller space, especially his combinatorial model.

While I am currently engaged in writing up my results, I can report a few. I have augmented the Masur-Minsky marking complex to obtain a space quasi-isometric to Teichmuller space in the Teichmuller metric (coincidentally and independently, Eskin-Masur-Rafi have built the same space and are using in their work on the rank of Teichmuller space). Using this graph and the Masur-Minsky hierarchy machinery, I am able to show that orbits of finite subgroups of the mapping class group have quasicenters in Teichmuller space, that every almost-fixed point is a uniformly bounded distance from a fixed point (which is unclear a priori given the positive curvature characteristics of the thin parts of Teichmuller space discovered by Minsky), and that almost-fixed points of these subgroups can be connected with paths that stay in the almost-fixed point set.

Camille Horbez

My research concentrates on the study of the group Out(Fn) of outer automorphisms of a ﬁnitely generated free group via its action on several geometric complexes, such as Culler and Vogtmann’s outer space CVn, or the complex of spheres. Understanding the topology of such Out(Fn)-complexes has indeed turned out to be a powerful tool to study algebraic properties of the group itself. I am particularly interested in understanding the topology at inﬁnity of these complexes.

37 38 6. SURFACE GROUPS, MAPPING CLASS GROUPS AND Out(Fn)

Outer space is defined to be the space of homothety classes of free, minimal, isometric actions of Fn on metric simplicial trees. A classical compactification of CVn was built in analogy to Thurston’s compactification of the Teichmller space of a surface. The translation length of an element g ∈ Fn in a tree T equipped with an Fn-action is defined as ||g||T := minx∈T d(x, gx). When the action is isometric, it only depends on the conjugacy class of the element g, so we get a map

i : CV → C(Fn) n PR , [T ] 7→ (||g||T )g∈C(Fn)

C(Fn) where C(Fn) denotes the set of conjugacy classes of elements of Fn, and PR is the corresponding projective space. Culler and Morgan have proved that the map i is injective, and actually an homeomorphism of CVn onto its image for a natural topology on CVn. Its image has compact closure, thus yields a compactification of CVn, which has been explicitly described by Cohen-Lustig and Bestvina-Feighn : it is the space of very small actions (i.e. actions with maximal cyclic arc stabilizers and trivial tripod stabilizers) of Fn on R-trees. I currently aim at understanding another compactification of CVn, defined via a notion of horofunctions for the Lipschitz (asymmetric) metric d on CVn. One motivation for studying this compactification comes from work by Karlsson and Ledrappier that established it as good framework for the study of random walks on CVn, whence on the group Out(Fn). Consider the map

ψ : CV → C(CV ) n n , z 7→ {x 7→ d(x, z) − d(b, z)} where C(CVn) denotes the space of real-valued functions on CVn, equipped with the topology of convergence on compact sets, and b is an arbitrary basepoint in CVn. This is an homeomorphism from CVn onto its image, and its image has compact closure by an Arzel-Ascoli type argument. So again, it defines a compactification of outer space. I aim at giving an explicit description of this compactification, and understand its topological properties, e.g. its homotopy type or its topological dimension. It turns out that this question is linked to the problem of strong spectral rigidity for the set of primitive elements of Fn : given an very small Fn-action on an R-tree T , to what extent is it characterized by the translation lengths of primitive elements of Fn ? I have also been working on a dual model of outer space coming from 3-dimensional topology : outer space can be defined to be the space of homotopy classes of sphere 1 2 systems in Mn := ]nS × S that divide Mn into punctured balls. Natural paths can be defined in this model of CVn via a surgery procedure, and I proved that these paths make definite progress in CVn when they remain in some thick part of CVn. This was done by relating the distance between two points in the thick part of outer space to the number of intersection circles between the dual spheres. Spheres can also be used to study the free splitting complex FSn of Fn, which has been proved by Handel and Mosher to be Gromov hyperbolic. In a joint work with A. Hilion, we gave a new proof of their result by understanding the geometry of surgery paths in a dual model of FSn defined in terms of spheres. Investigating the topology of FSn – e.g., finding analogues of Masur and Minsky’s tight geodesics in the curve complex, or understanding its Gromov boundary – seems to be a promising task in the study of the group Out(Fn). DAWID KIELAK 39

Chris Judge I am interested in the action of the mapping class group on various analytic spaces. For example, I study the stabilizer subgroups of totally geodesic surfaces in Teichm¨ullerspace. These stabilizer subgroups are essentially the automorphism groups of a natural 2-dimensional Euclidean polyhedral complex. The complex is essentially the moduli space of ellipses in a ﬂat surface with conical sigularities. I am also interested in the spectrum of the Laplacian for locally symmetric spaces. For example, I am interested in understanding the behavior of the spectra of hyperbolic 3-manifolds under coverings. A well-known—but in general, false—heuristic is that volume should be inversely related to the lowest nonzero eigenvalue of the Laplacian. Coverings provide a natural playground for testing when the heuristic and when it is false.

Dawid Kielak I am interested in various aspects of the theory of outer automorphisms of free groups (Out(Fn)), e.g., in the nature of the apparent similarities between these groups, lattices in semi-simple Lie groups (these include many arithmetic groups, e.g. GLn(Z)), and Mapping Class Groups (i.e. groups of homeomorphisms of a given surface modulo isotopy). One important property enjoyed by general linear groups over the integers is that ev- n m ery embedding Z ,→ Z (with m > n) induces an embedding GLn(Z) ,→ GLm(Z). An analogous statement is true for free groups and their automorphisms: writing Fm = Fn ∗ Fm−n allows us to construct Aut(Fn) ,→ Aut(Fm). The situation is far less clear, however, when we focus on groups of type Outn. Finding values of n and m for which there exist embeddings Out(Fn) ,→ Out(Fm) is not an easy task. We can also form a more general question, and ask about all possible homomorphisms Out(Fn) → Out(Fm). It is worth noting that this sort of question falls under the umbrella-term of rigidity. It is also worth noting that the understanding of such maps is crucial if we are to understand stability phenomena of various sorts (representation, (co)homological, etc.). I would also very much like to know whether Out(Fn) (for n > 3) has property FAb, that is if all its finite index subgroups have finite abelianisations. A group G enjoys property FAb for example if it has Kazhdan’s property (T). One class of groups with property (T) consists of the groups of type GLn(Z) for n > 2. It is an open and very interesting problem (with many important ramifications) if Out(Fn) has property (T). Proving that OutFn does not have property FAb would immediately tell us that it does not have (T); on the other hand proving that Out(Fn) has FAb would already imply many consequences of (T). The problem of having FAb or (T) is also open for the Mapping Class Groups. It is very likely that a proof showing that Out(Fn) does/does not have FAb/(T) can inspire (if not be generalised to) an analogous proof for the Mapping Class Groups. Recently I became also interested in the curvature properties of orthoscheme complexes of posets. In particular I would like to understand the curvature of the orthoscheme complex of the non-crossing partition complex – its geometry has im- plications for the geometry of braid groups (among other things). 40 6. SURFACE GROUPS, MAPPING CLASS GROUPS AND Out(Fn)

Thomas Koberda See page 62.

Sara Maloni My research interests lie at the intersection of hyperbolic geometry and low-dimensional topology. More precisely, I study Kleinian groups and, in particular, quasifuchsian groups, groups whose limit set is a topological circle. My work, up to this point, focuses mainly on the following topics: (1) S. Maloni, C. Series [2] Top terms of polynomial traces in Kra’s plumbing construction: Let Σ be a surface of negative Euler characteristic together with a pants decomposition P. Kra’s plumbing construction endows Σ with a projective structure which depends on some complex parameter τi. Let ρτ : π1(Σ) −→ PSL(2, C) be the associated holonomy representation. Then the traces of all elements ρτ (γ), with γ ∈ π1(Σ), are polynomials in the τi. We prove a formula giving a simple linear relationship between the coefficients of the top terms of ρτ (γ), as polynomials in the τi, and the Dehn–Thurston coordinates of γ relative to P. (2) S. Maloni [3] The asymptotic directions of pleating rays in the Maskit embedding: Applying the Top Terms’ Relationship, we determine the asymptotic directions of pleating rays in the Maskit embedding of a hyperbolic surface Σ as the bending measure of the ‘top’ surface in the boundary of the convex core tends to zero. Given a projective measured lamination [η] on Σ, the pleating ray P[η] is the set of groups in M for which the bending measure of the top component of the boundary of the convex core of the associated 3-manifold H3/G is in the class [η]. (3) S. Maloni [4] Slices of the quasifuchsian space: we define a new plumbing construction, called the c–construction, where c is a positive real vector. While Kra’s construction builds groups in the Maskit slice, the c– construction defines groups which lie on a slice of the quasi-Fuchsian space, called the Bers–Maskit slice which is a special connected component of the c–slice Lc(Σ) of the Quasifuchsian space. Other projects I’m working on now are: (1) Anti-de Sitter geometry: the globally hyperbolic maximal compact (GHMC) Anti-de Sitter (AdS) manifolds are analogous of Quasifuchsian groups in the Lorentzian geometry. In particular, I’m interested in the Bending Lamination Conjecture. (This is joint work with J-M Schlenker and J Danciger.) (2) Higher Teichmüller Spaces: I want to find an analogous of the Ahlfors– Bers theorem, of the Pleating Coordinates Theory and of the Grafting deformation for ‘Quasifuchsian groups’ in Lie groups of higher rank, that is Anosov representations ρ : π1(Σ) −→ PSL(n, C) which deform to Fuchsian representations. (3) character varieties: I’m studying the (relative) SL(n, C) character varieties of the four-holed sphere and the action of the mapping class group on the (relative) character variety, using a generalisation of Bowditch’s methods [1]. (This is joint work with F Palesi and S P Tan.) Finally, other projects I’m interested in are the following: EMILY STARK 41

(1) understand better the relationship between the plumbing parameters defined by Kra and by Earle–Marden; (2) investigate the slice of the quasifuchsian space of the once-punctured torus obtained by pinching an irrational geodesic lamination on the bottom surface; (3) extend the definition of the complex Fenchel–Niesen coordinates (defined for the quasifuchsian space QF(Σ)) to other subspaces of the representation variety R(Σ), for example the Schottky space S(Σ) ⊂ R(Σ); (4) investigate the relationship between the Fenchel–Nielsen coordinates and Bonahon’s shearing and bending coordinates; (5) prove a Schläfliformula for the Chern–Simons invariants of a hyperbolic 3–manifold; References [1] B. Bowditch Markoff triples and quasi-Fuchsian groups, Proc. London Math. Soc. 77 (3) (1998), 697-736. [2] S. Maloni, C. Series Top terms of polynomial traces in Kra’s plumbing construction, Algebraic and Geometric Topology 10 (3) (2010), 1565–1607. [3] S. Maloni The asymptotic directions of pleating rays in the Maskit embedding, submitted. [4] S. Maloni Slices of the quasifuchsian space, preprint. [5] S. Maloni Wrapping and self-bumping, preprint.

Emily Stark I am interested in questions in geometric group theory and low-dimensional topology, with a focus on cube complexes and amalgamated products of hyperbolic surface groups. To understand the structure of hyperbolic surface group amalgams, we consider their classification up to (abstract) commensurability. Closed hyperbolic surface groups are commensurable; every closed hyperbolic surface covers the genus two surface. However, commensurability of amalgamated products of hyperbolic surface groups over Z depends on the monomorphisms from the Z subgroup into each surface group. We show that if all maps send a generator to the homotopy class of a simple closed curve, then the commensurability classification depends only on the ratio of the Euler characteristic of the surfaces and on the homology of the image. I hope to understand the commensurability classes when the monomorphisms map the generator of the common Z subgroup to immersed curves. I am also interested in the interplay between CAT(0) cubed geometry and hyperbolic geometry, a problem I am considering in the context of hyperbolic surface group amalgams. One focus of my current research is on the distinction between the geometric dimension and the CAT(0) cubed dimension of a group, where the geometric dimension is the minimal dimension of a K(G, 1) space for the group, and the CAT(0) cubed dimension is the minimal dimension of a CAT(0) cube complex on which the group acts geometrically. Hyperbolic surface group amalgams have geometric dimension equal to 2, and they are known to act geometrically on finite dimensional CAT(0) cube complexes by extending the work of Sageev, or by applying theorems of Wise [1], [2]. The CAT(0) cubed dimension of these groups is equal to 2 when a generator of the common Z subgroup is identified with simple closed 42 6. SURFACE GROUPS, MAPPING CLASS GROUPS AND Out(Fn) curves; I am working to understand the dimension for amalgams over immersed curves. References [1] M. Sageev, “Ends of group pairs and non-positively curved cube complexes.” Proc. London Math. Soc. (3) 71, pp. 585-617. 1995. [2] D. Wise, “The structure of groups with a quasiconvex hierarchy.” Preprint, 2011.

Vera Toni´c See page 19. CHAPTER 7

Three manifold groups, knots and braids

Moshe Cohen I present a relatively new combinatorial model for knots called the balanced over- laid Tait graph, developed in a string of my papers with co-authors. This model translates familiar elements in knot theory to perfect matchings, or dimer coverings, which appear very often in the literature of ﬁelds like graph theory and even statistical physics. Some current applications of this model in my work include: dimer models for the Jones polynomial of pretzel knots, the Alexander polynomial, and the twisted Alexander polynomial; the reformulation of Kauﬀman’s clock lattice as a graph of perfect matchings, with applications to the study of grid graphs via harmonic knots as well as the study of discrete Morse functions via complexes; and dimer models for knot homology theories, begun in and extending to future work.

There is a bijection between the set of all knot or link diagrams and the set of all signed plane graphs G. Spanning tree expansions of G have been used to produce several models of use in knot theory. There is a another bijection between the set of (rooted) spanning trees (or arborescences) of a plane graph G and the set of perfect matchings (or dimer coverings) of a related plane bipartite graph Γ that has been explored in my previous work, as well as in work by Kenyon, Propp, and Wilson, who discuss its origins. The theorem and its proof are a generalization of a result of Temperley (1974) which is discussed in problem 4.30 of a 1979 book by Lov´asz. The related graph Γb also appeared in work by Huggett, Moﬀatt, and Virdee. The graph Γ is currently being studied by Kravchenko and Polyak for knots on a torus in relation to cluster algebras. Dimers themselves, developed by Kasteleyn and Temperley-Fisher in the 1960’s as a tool for studying statistical physics, have been studied extensively, as well; see lecture notes by Kenyon. By weighting the edges of a graph appropriately and summing over all perfect matchings µ and all edges ε in the perfect matching, one hopes to obtain a dimer model for a particular invariant.

Theorem:The graph Γ gives a dimer model for the Jones polynomial of pretzel knots and the Alexander polynomial of any knot. An extended version of Γ gives a dimer model for the twisted Alexander polynomial of any knot.

Discrete Morse theory was introduced in 1995 by Forman to apply the power of the classical version to combinatorially deﬁned complexes. The critical points found in the smooth version can be determined combinatorially by collapsing pairs of i- and i + 1-dimensional cells in the complex C, and these collapses are described (and ordered) using the following map.

43 44 7. THREE MANIFOLD GROUPS, KNOTS AND BRAIDS

A discrete Morse function is a weakly increasing map f :(C, ⊆) → (Z, ≤) such that |f −1(n)| ≤ 2 for all n ∈ Z and such that f(σ) = f(τ) implies that one of σ, τ is a face of the other. A critical cell for a discrete Morse function f is face of C at which f is 1-to-1.

Proposition: Perfect matchings on Γ constructed from a knot diagram D correspond to discrete Morse functions on the 2-complex ∆ of the 2-sphere whose 1-skeleton is the Tait graph G. In future work, I plan to use properties of the perfect matching model for knots to explore the space of discrete Morse functions, beginning with knot diagrams as quadrangulations of the 2-sphere. Notions from this theory can be used to generalize the perfect matching model for knots on higher-genus surfaces and even for knotted surfaces in four dimensions.

Shelley Koone My current research project is an extension of a paper by Yo’av Rieck and Tsuyoshi Kobayashi, cf. http://arxiv.org/abs/0812.4476. I am trying to generalize thier construction of a manifold that admits both strongly irreducible and weakly reducible minimal genus Heegard splittings to higher genus Prior to beginning work on my dissertation I studied various aspects of knot theory.

Piotr Przytycki Here is a brief description of our recent joint work with Daniel T. Wise on separability of embedded surfaces in 3–manifolds. A subgroup H ⊂ G is separable if H equals the intersection of finite index subgroups of G containing H. Scott proved that if G = π1M for a manifold M with universal cover Mf, then H is separable if and only if each compact subset of H\Mf embeds in an intermediate finite cover of M (Lem 1.4 [5]). Thus, if H = π1S for a compact surface S ⊂ H\Mf, then separability of H implies that S embeds in a finite cover of M. Rubinstein–Wang found a properly immersed π1–injective surface S # M in a graph manifold such that S does not lift to an embedding in a finite cover of M, and they deduced that π1S ⊂ π1M is not separable (Ex 2.6 [4]). Our main result is: Theorem 1. Let M be a compact connected 3–manifold and let S ⊂ M be a properly embedded connected π1–injective surface. Then π1S is separable in π1M. The problem of separability of an embedded surface subgroup was raised for instance by Silver–Williams — see [6] and the references therein to their earlier works. The Silver–Williams conjecture was resolved recently by Friedl–Vidussi in [2], who proved that π1S can be separated from some element in [π1M, π1M] − π1S when- ever π1S is not a fiber. We proved Theorem 1 when M is a graph manifold in Thm 1.1 [3]. Theorem 1 was also proven when M is hyperbolic [7]. In fact, every finitely generated subgroup of π1M is separable for hyperbolic M, by [7] in the case ∂M 6= ∅ and by Agol’s theorem [1] for M closed. References [1] I. Agol, D. Groves, J. Manning, The virtual Haken conjecture, (2012) arXiv:1204.2810 PIOTR PRZYTYCKI 45

[2] S. Friedl, S. Vidussi, A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4–manifolds, (2012), arXiv:1205.2434 [3] P. Przytycki, D.T. Wise, Graph manifolds with boundary are virtually special, submitted, (2011) arXiv:1110.3513 [4] J.H. Rubinstein, S. Wang, π1–injective surfaces in graph manifolds, Comment. Math. Helv., 73 (4) (1998), 499–515 [5] P.Scott, Subgroups of surface groups are almost geometric, J. London Math. Soc. (2), 17 (3) (1978), 555–565 [6] D.S Silver, S.G. Williams, Twisted Alexander polynomials and representation shifts, Bull. Lond. Math. Soc. 41 (3) (2009), 535–540 [7] D.T. Wise, The structure of groups with quasiconvex hierarchy, 2011, submitted, http://www.math.mcgill.ca/wise/papers.html

CHAPTER 8

Lattices in locally compact groups

Amichai Eisenmann I am interested in general in the theory of infinite groups. More particularly here are some issues I dealt with in the past: My Ph.D. thesis was concerned with the counting of arithmetic lattices of bounded co-volume in G := PSL2(K) for a p-adic field K. More precisely, fixing a Haar measure on G and given x > 0 there is a finite number of arithmetic lattices of co-volume bounded by x. The main result obtained in my thesis was that, letting ALG(x) denote the number of lattices of ALG(x) co-volume bounded by x, the limit limx→∞ x log x exists, (and may be computed) for a large family of p-adic fields. This result is an analogue to a previous result of a similar nature for the group PSL2(R).

More recently there is a result (joint with N. Monod) concerning normal generation of locally compact groups. There is an open question of J. Wiegold, whether a finitely generated (discrete) perfect group is necessarily normally generated by a single element. It is widely believed the answer should be negative. We considered the analogous question for locally compact groups, assuming they have no infinite discrete quotients (which would bring us back to Wiegold’s question). It turns out that here the answer is positive. More precisely: If G is a locally compact compactly generated perfect group without infinite discrete quotients, then G is topologically normally generated by a single element.

Another work (again joint with N. Monod) is concerned with CAT(0) lattices. The question addressed was whether the famous (non-residually ﬁnite) Baumslag Soli- tar groups < x, y|xynx−1 = ym > appear in CAT(0) lattices. It turns out that on the one hand the Baumslag-Solitar groups cannot be themselves CAT(0) lattices (even non-uniform), but on the other hand they can appear as subgroups of CAT(0) lattices. We give explicit construction of these lattices as some extensions of arithmetic groups by free inﬁnitely generated groups.

Currently I am interested in the action of Aut(Fn) on the presentation variety of the free group in p-adic groups. This is ongoing work with Y. Glasner, and is yet in a preliminary stage. The object is to generalize to P GLn(F ) a previous result of Y. Glasner, stating that the action of Aut(Fn)(n ≥ 3) by precomposition on the set {h : Fn → P GL2(F )|Im(h) = P GL2(F )}, for a p-adic field F , is ergodic. In addition I have become interested lately in the study of profinite properties of groups, i.e. finiding out when various group properties depend only on its finite quotients. Here also work is still in a preliminary stage.

47 48 8. LATTICES IN LOCALLY COMPACT GROUPS

Swiatoslaw Gal See page 74.

Chris Judge See page 39.

Sanghoon Kwon My research interest lies in both geometric group theory and homogeneous dynamics. In particular, I am currently interested in problems about automorphism groups of polyhedral complexes. One of my research topic is a non-Archimedean version of the hyperbolic lattice point problem. We can view it as a problem on automorphism groups of simplicial complexes and their discrete subgroups. The lattice point problem in hyperbolic spaces is the following question: For a + n discrete subgroup Γ of Isom (H ), what is the asymptotic of NT (x, y, Γ) = #{γ ∈ Γ: d(x, γy) ≤ T }? It is a generalization of the Gauss circle problem in the Euclidean plane. There are results about this problem when Γ is cofinite in Lie group G over various fields. This appeared in Effective equidistribution of S-integral points on symmetric varieties, by Yves Benoist and Hee Oh. Also, when Γ is a geometrically finite group of Isom+(Hn), the asymptotic is found in varius article; for instance, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces, by Peter Lax and Ralph Phillips. Lattice point problem has many arithmetic applications. Likewise, counting orbit of thin groups can be applied to the problems in number theory; for example, Appolonian circle packings and closed horospheres on hyperbolic 3-manifolds, by Alex Kontorovich and Hee Oh. My research is now focused on a thin group Γ of Lie group G over non-Archimedean local fields, which is a closed subgroup of automorphism group of simplicial complexes. More specifically, G acts on a Bruhat- Tits building. I would like to find an asymptotic of orbit with nice error term.

Henning Niesdroy Recently I started my Ph.D. studies under supervision of Prof. Dr. Kai-Uwe Bux. Before I graduated, I was particularly interested in topology and group theory. In order to combine these two topics I joined the group of my supervisor to work in geometric group theory. In the following, I give a short introduction of what my thesis is about: n Consider SLn(R), the hyperbolic space H := SLn(R)/SOn(R) and the action of n n SLn(Z) on H . Reduction theory describes a fundamental domain Sn ⊂ H . Generalizing reduction theory to S-arithmetic groups, Godement found an adelic formulation treating all places simultaneously. Let K be a global number field, G be a reductive group (think of SLn), and let A be the ring of adeles of K. Then G(K) is discrete in G(A), and Godement finds a fundamental domain (coarsly) for the action of G(K) on G(A). Later, Behr and Harder transferred this to the case when K is not a global number field, but a function field. In 2012 Bux-Köhl-Witzelgave a geometric reformulation of Behr-Harder. Now the natural question arises, whether this geometric reformulation can be transferred back to the case of a number field. My task is to actually do that. HENNING NIESDROY 49

Further we just started a new project in our group. The question is, whether −1 SL2(Z[t, t ]) is ﬁnitely generated or not. This question has been open for more than thirty years, though I cannot tell who came up with it ﬁrst.

CHAPTER 9

Buildings, Coxeter groups and Artin groups

For right-angled Coxeter/Artin groups see also Chapter 10.

Antoine Clais A right-angled hyperbolic building is a cellular complex made of n-cells, named chambers, isometric to a right-angled compact polyhedron P of the standard hyperbolic space Hn. Moreover this complex admits a subfamily of cellular complexes, called apartments, isometric to the tilling of Hn by P .

We would like to establish the Combinatorial Loewner Property (CLP) for the boundary of such a building. This property may be deﬁned has follow.

First we deﬁne the Combinatorial Modulus which is essentially a measure (in a weak sense) on the set of all the curves in ∂X. Then, denoting by A and B two connected compact and disjoint subsets of ∂X, we say that the (CLP) is satisﬁed if the Combinatorial Modulus of the set of all curves joining A to B is controlled by the relative distance between A and B : dist(A, B) ∆(A, B) = . min{diam(A), diam(B)} This might be true because of the large amount of symmetries that the action of Γ induces on the curves of ∂X.

The (CLP) have been established recently by Bourdon and Kleiner for the boundary of hyperbolic Coxeter groups and then used by them to ﬁnd a new proof of the Cannon conjecture for Coxeter groups. Finding the CLP at the boundary of some hyperbolic buildings could lead to some rigidity results on the buildings.

Kamil Duszenko My current research is focused on Coxeter groups and their actions on CAT(0) and CAT(−1) spaces. Recall that a Coxeter group is a group given by the presentation mst hS|{(st) = 1}s,t∈Si, where S is a finite set and mss = 1, mst ∈ {2, 3,..., ∞} for s 6= t. To be more specific, I would like to determine the smallest possible dimension of a CAT(0) space on which a given Coxeter group can act without a global fixed point. Also, I am addressing the question if every non-affine Coxeter group acts nontrivially on a CAT(−1) space, or even admits a non-elementary Gromov-hyperbolic quotient. Every Coxeter group acts on the CAT(0) Davis complex, however, its dimension is usually far from being optimal. On the other hand, the existence of an action on a CAT(−1) space is known only in some specific cases. The dimension

51 52 9. BUILDINGS, COXETER GROUPS AND ARTIN GROUPS of such a CAT(−1) space can be bounded from below by a number related to the rank of a minimal non-affine special subgroup. I have shown, among other things, that all minimal non-affine Coxeter groups can be mapped surjectively onto non-elementary groups acting geometrically on CAT(−1) spaces (in particular, these quotients are Gromov-hyperbolic). As an example how such results can be applied, let us mention that the above result led to a proof that all non-affine Coxeter groups have unbounded reflection length, where the reflection length of an element w of a Coxeter group (W, S) is the smallest n such that w is equal to the product of n reflections, i.e. elements of the form vsv−1, where v ∈ W and s ∈ S. References [1] K. Duszenko, Reflection length in non-affine Coxeter groups, Bull. London Math. Soc. (2012) 44(3):563-570.

Thomas Haettel I am interested in understanding the asymptotic geometry of homogeneous spaces, such as the space of maximal ﬂats of a symmetric space of non-compact type or of a Euclidean building. I am also interested in CAT(0) spaces and groups. With Dawid Kielak and Petra Schwer, we have been recently studying simplical K(π, 1)’s for braid groups and other Artin groups of ﬁnite type, described by Tom Brady, Jon McCammond and Colum Watt. We have proved that some of them are CAT(0).

Riikka Kangaslampi See page 24.

Sang-Jin Lee I have studied the conjugacy problem in the Garside groups. Garside group is a lattice-theoretic generalization of finite type Artin groups. I am interested in general Garside groups, but also concerned with algebraic and geometric properties of particular Garside groups such as the Artin groups of finite type and the braid groups associated to the complex reflection groups. Recently, Eon-Kyung Lee and I investigated the discreteness of the translation numbers in Garside groups, showing that, in terms of the properties of translation numbers, Garside groups are as good as word hyperbolic groups [1, 2]. We also studied some related topics in the braid groups and the Artin groups [6, 7, 8]. I am also interested in the geometric group theoretic approaches to the braid groups and the Artin groups of finite type such as [1, 2, 3]. References [1] M. Bestvina, Non-positively curved aspects of Artin groups of finite type, Geom- etry and Topology 3 (1999) 269–302. [2] T. Brady and J. McCammond, Braids, posets and orthoschemes, Algebr. Geom. Topol. 10 (2010), no. 4, 2277–2314. [3] J. Crisp and L. Paoluzzi, On the classification of CAT(0) structures for the 4-string braid group, Michigan Math. J. 53 (2005) 133–163. [4] E.-K. Lee and S. J. Lee, Translation numbers in a Garside group are rational with uniformly bounded denominators, J. Pure Appl. Algebra 211 (2007) no. 3, 732–743. PETRA SCHWER 53

[5] E.-K. Lee and S. J. Lee, Some power of an element in a Garside group is conjugate to a periodically geodesic element, Bull. Lond. Math. Soc. 40 (2008) no. 4, 593–603. [6] E.-K. Lee and S. J. Lee, A Garside-theoretic approach to the reducibility problem in braid groups, J. Algebra 320 (2008) no. 2, 783–820. [7] E.-K. Lee and S.-J. Lee, Uniqueness of roots up to conjugacy for some aﬃne and ﬁnite type Artin groups, Math. Zeit. 256 (2010), no. 3, 571–587. [8] E.-K. Lee and S.-J. Lee, Injectivity on the set of conjugacy classes of some monomorphisms between Artin groups, J. Algebra 323 (2010), no. 7, 1879–1907.

Petra Schwer See page 34.

CHAPTER 10

Groups acting on cubical complexes and spaces with walls

Yago Antol´ınPichel My current research is the study of inﬁnite discrete groups from a combinatorial and geometrical point of view and its connections with low-dimensional topology. In general, I like to study groups from diﬀerent perspectives. In this statement I will discuss some results about two families of groups on which I have worked most: right angled Artin groups and one-relator groups.

Right angled Artin groups. Right angled Artin groups are special examples of graph products. Let Γ be a simplicial graph and suppose that G = {Gv | v ∈ V Γ} is a collection of groups (called vertex groups). The graph product ΓG, of this collection of groups with respect to Γ, is the group obtained from the free product of the Gv, v ∈ V Γ, by adding the relations

[gv, gu] = 1 for all gv ∈ Gv, gu ∈ Gu such that {u, v} is an edge of Γ. A graph product of infinite cyclic groups is a right angled Artin group (raag for short). The family of finitely generated (virtually) subgroups of raags is very rich including, after the work of Wise [16] and Agol [1], most 3-manifold groups. In [10], Minasyan and I have shown that if a family of groups G satisfies a Tits alternative, then the group ΓG also satisfies this alternative. In particular, we found a strong criterium for a group being a subgroup of right angled Artin group: Theorem ([10, Corollary 1.6]). Any non-abelian subgroup of a right angled Artin group maps onto F2, the free group of rank two. Recently, with Aditi Kar, we have studied one-relator quotients of graph products. In particular, we focus our attention to the case of right angled Artin groups. A graph is starred if the defining graph contains no line of length 4 or a square as a full subgraph. In [8] we prove

Theorem ([8, Theorem B]). Let Γ be a starred graph and G = {Gv | v ∈ V Γ} be a family of poly-(inﬁnite cyclic) groups. Let g ∈ G = ΓG. Then, the word problem of the one-relator quotient G/hgi is solvable. In [8] we also develop a Freiheitssatz for the case of one-relator quotients of right angled Artin groups over starred graphs. Both results follow for a Magnu’s-like induction technique for free products amalgamated over a central subgroup. The growth of a group has been proved to be a powerful tool to detect algebraic properties of groups. The geodesic growth is much more delicate, and detects

55 56 10. GROUPS ACTING ON CUBICAL COMPLEXES AND SPACES WITH WALLS properties of a group with respect to a generating set. The geodesic growth series is given by X t|w| w∈L(X) where L(X) is the language of geodesic words in a group over the generating set X and |w| is the word length of w. Question ([13, Question 1]). Is the geodesic growth series (with respect to the standard generating sets) are a complete invariant for raags? In [6], together with Ciobanu, we give a negative answer to this question, giving examples of non-isomorphic raags with the same geodesic growth series. The result is a Corollary of a more general statement about Coxeter groups.

One-relator groups. I am interested on one-relator groups as examples of groups of cohomological dimension 2. Great part of my research has been devoted to understand classyfying spaces, L2-homology or Bredon Cohomology of generalizations of one-relator groups. For example, in my Phd thesis, I introduced a family of two relator groups, called Hempel groups, in order to study surface-plus-one relation groups (which where studied by Hempel in [12]). To be more precise, Hempel groups is a family of two-relator groups of the form,

(10.1) hx, y, z1, . . . , zn|[x, y] = w(z1, . . . , zn), r(x, y, z1, . . . , zn) = 1i ±1 where [x, y] is the commutator of x and y, w(z1, . . . , zn) is a word in {z1, . . . , zn} , ±1 and r(x, y, z1, . . . , zn) is any word in {x, y, z1, . . . , zn} . There are some extra technical assumptions on r, however we ignore them for the sake of simplicity. For example, let G be a Hempel group given as in (10.1), let Cr be the cyclic subgroup generated by the root of r in hx, y, z1, . . . , zn|[x, y] = wi. Some of the properties obtained in my work with Dicks and Linnell [3], are the following

Theorem ([3, Theorem 6.5 and 7.3]). Let G and Cr as above then (i) Every torsion-free subgroup of G is locally indicable. (ii) G is virtually torsion-free. (iii) The following sequence of ZG-modules n+2 0 → ZG ⊕ Z[G/Cr] → ⊕i=1 G → ZG → Z → 0 is exact. The algebraic properties of Hempel groups are used in [3] to compute the L2- Betti numbers of (non-orientable) surface-plus-one-relation group. (See [15] for the definition of L2-Betti numbers). Theorem ([3, Theorem 9.5]). Let G be surface-plus-one-relation group. Then G is of type VFL and, for each n ∈ N ,  1 max{χ(G), 0} = |G| if n = 0, (2)  βn (G) = max{−χ(G), 0} if n = 1, 0 if n ≥ 2. In [7], toguether with Flores, we study classifying spaces for proper actions for groups (see [14] for definitions and motivation) with the following property YAGO ANTOLÍN PICHEL 57

(C) There is a finite family of non-trivial finite cyclic subgroups {Gλ}λ∈Λ such that for each non-trivial torsion subgroup H of G, there exists a unique −1 λ ∈ Λ and a unique coset gGλ ∈ G/Gλ such that H 6 gGλg . Groups satisfying (C) have a very controlled torsion. One-relator groups or more generally Hempel groups are examples of groups satisfying (C). This classifying spaces allow us to compute, in [7], the Bredon Homology of Hempel groups with coefficients in the complex representation ring, and is used to describe the K-theory of the reduced C∗-algebra of Hempel groups via the Baum-Connes assembly map.

Other work and current research. Some work that I have not discussed include the study of virtually free groups [2], generalizing the recent proof of the Hanna-Neumman conjecture to free products [9], or understanding the asymptotic behaviour of homogeneous equations in surface groups [5]. At the present moment, I am studying subgroups of raags that have hyperbolically embedded subgroups (in the sense of [11]). References [1] Ian Agol, The virtual Haken conjecture, Preprint (2012) [2] Yago Antol´ın, On Cayley graphs of virtually free groups. Groups Complex. Cryptol, 3 (2) (2011), 301 – 327. [3] Yago Antol´ın,Warren Dicks and Peter A. Linnell, Non-orientable surface-plus- one-relation groups, J. Algebra, 326 (2011), 4–33. [4] Yago Antol´ın,Warren Dicks and Peter A. Linnell, On the Cohen-Lyndon Local- Indicability Theorem, Glasgow Math. J., 53 (2011), 637– 656. [5] Yago Antol´ın,Laura Ciobanu and Noelia Viles, On the asymptotics of visible elements and homogeneous equations in surface groups, Groups. Geom. Dyn (To appear) [6] Yago Antol´ın and Laura Ciobanu, Geodesic growth in right-angled and even Coxeter groups, Submitted to publication. [7] Yago Antol´ınand RamónFlores, On the classifying space for proper actions of groups with cyclic torsion, Forum Math. (to appear) [8] Yago Antol´ınand Aditi Kar, One Relator Quotientes of Graphs Products, Sub- mitted to publication. [9] Yago Antol´ın,Armando Martino and Inga Schwabrow, Kurosh rank of intersec- tions of subgroups of free products of orderable groups, Submitted to publication. [10] Yago Antol´ınand Ashot Minasyan, Tits alternatives for graph products, Sub- mitted to publication. [11] Fran¸cois Dahmani, Vincent Guirardel and Denis Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces Preprint 2011 [12] John Hempel, One-relator surface groups, Math. Proc. Cambridge Philos. Soc. 108(1990), 467–474. [13] J. Loeffler, J. Meier and J. Worthington, Graph products and Cannon pairs, Internat. J. Algebra Comput. 12 (2002), 6, 747–754. [14] Wolfgang Lueck, Survey on classifying spaces for families of subgroups, in “In- finite Groups: Geometric, Combinatorial and Dynamical Aspects”, Progress in Mathematics, Vol. 248, Birkhäuser(2005) [15] Wolfgang Lück, L2-invariants: theory and applications to geometry and K- theory, Ergeb. Math. Grenzgeb.(3) 44, Springer-Verlag, Berlin, 2002. 58 10. GROUPS ACTING ON CUBICAL COMPLEXES AND SPACES WITH WALLS

[16] Daniel Wise, The structure of groups with a quasiconvex hierarchy, Electronic Research Announcements in Mathematical Sciences (2009).

Sylvain Arnt My field of research is Geometric Group Theory; more precisely, I study isometric affine actions of groups on some normed spaces of function, like the Haagerup property (a-(T)-menability) or the property (T) of Kazhdan. My work is essentially based on a result of G.Yu which says that, for a finitely generated word-hyperbolic group G, there exists an isometric affine and proper action of this group on a `p space i.e. G is a-FLp-menable. I want to generalize this theorem to a larger class of groups, containing hyperbolic groups and which is stable by direct, free, amalgamated products. In order to do that, we can see in the proof of the theorem of G.Yu that the cocycle provided for the affine proper action gives a good way to separate points in the space G in the sense of walled spaces. Recall that a wall space is a space X together with a collection W of partitions in two pieces of X called “walls“ and the condition that for two fixed points in X, there is finitely many walls such that one point is contained in a piece (called halfspace) of the wall and the other point is contained in the other halspace. If this case happens, we say that the wall separates this two points. We can weaken this definition when we can define a measure on the set of walls, saying that (X,W ) is a space with measured walls if the measure of the set of walls separating any two points of X is finite. The main results over this notion which interest me in my work are the following: Theorem 1: If G is a locally compact and second countable, then G has the Haagerup property if and only if, it admits a proper continous action by automorphisms on a space with measured walls. Theorem 2: If G is finitely generated acting properly on a space with walls, then there exists a CAT(0) cube complex on which G acts properly.

Using the ideas of space with walls and the notion of separation of points in hyperbolic groups, we can deﬁne the structure of space with ”rooms”: Deﬁnition: Let X,M be sets and A(M) be a normed space of real valued functions on M. We say (X,M,A(M)) is a space with rooms if there exists a function f : M × X × X → [0, 1] such that: - for all m ∈ M, (x, y) 7→ f(m, x, y) is a pseudo-distance on X, - for all x, y ∈ X, m 7→ f(m, x, y) belongs to A(M).

We say that a finitely generated group G together with a finite set of generators S is a roomed group (w.r.t. S) if we can define a structure of roomed space on X = Cay(G, S). Thanks to the theorem of Yu, we can show that hyperbolic groups are roomed spaces; moreover, I proved that the set of roomed groups is stable by direct and free product and, under additionnal conditions on the function f : M × G × G → [0, 1], my work in progress is to show that a roomed group (G, M, A(M)) acts isometrically properly on the affine space A(M) which general- izes theorem 1’s necessary condition. I also want to have a build a similar complex as in theorem 2 on which a roomed group acts properly. MARK HAGEN 59

Another field that interests me is metrizability of group. We already have a good overview about metrizability of topological groups: Birkhoff and Kakutani showed that a topological group is metrizable if and only if it’s Hausdorff and there exists a countable fundamental system of neighbourhoods of the identity element. And in this case the topology can be described by a left-(or right-)invariant metric. But the distance constructed to prove this fact, just cares about local in- formations of the group and need not be a proper metric. As we can see in a paper of Lubotzky, Mozes, and Ragunathan in the case of compactly generated, second countable groups and in a paper of Haagerup and Przybyszewska in the case of locally compact, second countable groups, given a left-invariant distance which generates the topology, we can construct a new metric which is plig (proper, left-invariant, generates the topology); futhermore, the previous result was already established; in one of his paper, Struble showed the following theorem: a locally compact group G admits a plig metric if, and only if G is second countable. The previous metrics contructed to prove these facts are not really computable. My work on this subject was to define an explicit plig metric on locally compact and compactly generated groups, which depends only on the Haar measure on the group and the choice of a compact generating set.

Radhika Gupta I am a second year graduate student at the University of Utah. My interests are centered around geometry. Currently I am exploring different research areas to identify a suitable one. I am familiarizing myself with the field of geometric group theory through a reading course. I am reading about CAT(0) cube complexes. I have read M.Sageev’s construction of a cubing form a multi-ended pair (G, H) and that there is a natural essential action of G on this cubing. I studied RAAGs, Coxeter groups and that Coxeter Groups are special in a paper by F.Haglund and D.T.Wise. I also read about the action of Coxeter Groups on cube complexes in a paper by G.A.Niblo and L.D.Reeves. I think the Young Geometric Group Theory Seminar will be an excellent opportunity for me to explore the field of geometric group theory and give me a broad perspective of the field as I am starting work in it.

Mark Hagen My research concerns CAT(0) cube complexes, their geometric properties, and the question of which groups do and do not act nicely on these spaces. At the moment, I’m particularly interested in a few related large-scale features of a CAT(0) cube complex X that admits a proper, cocompact action by a group G. In general, two mutually exclusive properties that a group G can have are: relative hyperbolicity (here, {G} doesn’t count as a set of peripheral subgroups) and thickness. Roughly, a space is thick of order 0 if it has linear divergence function (e.g. Euclidean space in dimension at least 2), and thick of order at most n if it is the coarse union of quasiconvex subspaces, each thick of order at most n − 1, such that the following graph Λ is connected: Λ has a vertex for each of the given subspaces, with two vertices adjacent if the corresponding subspaces have unbounded, path-connected coarse intersection. The property of being thick (i.e. having well-defined order of 60 10. GROUPS ACTING ON CUBICAL COMPLEXES AND SPACES WITH WALLS thickness) is enjoyed by many cubical groups, including all one-ended right-angled Artin groups, as well as many groups that do not act nicely on cube complexes. Jason Behrstock and I have recently characterized CAT(0) cube complexes whose asymptotic cones do not have cut-points, and used this to show that both relative hyperbolicity and thickness of G can be detected (and, indeed, characterized) by looking at the induced action of G on the simplicial boundary of X, which is a combinatorial analogue of the Tits boundary, defined in terms of the hyperplanes and halfspaces in X. It is unknown at present what can be said if the cocompactness hypothesis is relaxed, and whether such results can be used to prove that specific groups that are known to be thick (or relatively hyperbolic) and cubulated (i.e. act metrically properly on cube complexes) do not act properly and cocompactly on cube complexes. It seems that, in the absence of some amount of hyperbolicity, cocompact cubulation is not often achieved; a recent result of Wise describing the tubular groups – which are thick – that can be cubulated, and the much more restricted class of tubular groups that can be cocompactly cubulate, seems to bear this out, and it would be interesting to see more examples of criteria, for specific classes of cubulated groups, that preclude cocompact cubulation. Another interesting question deals with divergence functions of cubulated groups. The definition of a thick metric space is designed to study divergence, and Behrstock- Drut¸u showed that a related invariant, the order of strong thickness, provides a polynomial upper bound on the divergence function of a metric space. It is not known for which spaces the converse, i.e. that a space of polynomial divergence function is strongly thick, holds, but this question seems to be approachable for proper, cocompact CAT(0) cube complexes. I’ve also lately been interested in the question of which median groups are cubical (or even cubulated). More precisely, if the group G acts on a median space without a global fixed point, under what additional hypotheses must G act nontrivially on a CAT(0) cube complex? This question is particularly interesting in light of the fact, due to Chatterji-Drut¸u-Haglund, that every action of G on a median space has bounded orbits if and only if G has property (T).

Sang-hyun Kim My research area is Geometric Group Theory and Low-Dimensional Topology. The following question, due to Gromov, has intrigued me into a couple of past and ongoing projects: does every one-ended word-hyperbolic group contain a closed hyperbolic surface subgroup? More specifically, I study hyperbolic surface subgroups of right-angled Artin groups and other CAT(0) groups. I discovered an infinite family of right-angled Artin groups which contain a hyperbolic surface subgroups while their defining graph do not have long (≥ 5) cycles as an induced subgraph; this answered a question due to Gordon–Long–Reid. Also, there is a decomposition theorem for the graphs which define right-angled Artin groups without hyperbolic surface subgroups. I also consider a CAT(0) graph of free groups with cyclic edge groups. The double of a 2-dimensional CW-complx X is obtained by taking two copies of X, puncturing each of the 2-cells, and gluing along the holes thus obtained. The double may be considered as the “connected sum” of two copies of X. With Sang-il Oum (KAIST), I proved that the fundamental group of the double of any one-ended two-generator presentation complex contains a hyperbolic surface group. This depends on a tool, THOMAS KOBERDA 61 called polygonality, of words in free groups defined by Henry Wilton (University College London) and me. Another theme of my study is embedability between groups. Using mapping class groups, Thomas Koberda (Harvard University) and I studied embeddings between right-angled Artin groups. We combinatorially characterized right-angled Artin subgroups of a given right-angled Artin group when the latter was defined by either a triangle-free graph or a forest. In particular, we completely determined when there is an embedding between right-angled Artin groups on cycles. We also proved that the chromatic number of the defining graph is an obstruction to embedding a right-angled Artin group into a mapping class group. Key ingredients of the proof are realization of right-angled Artin groups as subgroups of mapping class groups and studying the group elements corresponding to pseudo-Anosov elements. Our discovery is the extension graph, which “explains” all the right-angled Artin subgroups of a given right-angled Artin group. Koberda and I are working on a project on hyperbolic aspects of right-angled Artin groups, in relation to their actions on extension graphs.

Thomas Koberda I have been interested in the algebraic and geometric structure of right-angled Artin groups. Recall that a right-angled Artin group is deﬁned by a ﬁnite graph Γ. The generators of A(Γ) are the vertices of Γ, and edges of Γ impose commutation relations. To a graph Γ, one can associate the extension graph Γe whose vertices are given by vg, where v is a vertex of Γ and where g ∈ A(Γ), and where edges are given by commutation in A(Γ). Together with S. Kim, we have been developing an analogy between right-angled Artin groups and mapping class groups, via their respective canonical actions on the extension graph and the curve complex. The most fruitful method for developing this analogy is to embed right-angled Artin groups into mapping class groups, and then to use the machinery which already exists there. For instance, combining the results of [2], [3] and [5], one can determine precisely the right-angled Artin groups which embed into a given mapping class group and into a given right-angled Artin group, using the structure of the curve complex and extension graph respectively. The analogy runs much deeper, however. It turns out one can generalize much of the Masur–Minsky machinery of [6] and [7] to right-angled Artin groups, thus obtaining results like: • The extension graph is Gromov hyperbolic. • The action of the right-angled Artin group on its extension graph is acylindrical (cf. [1]). • A Bounded Geodesic Image Theorem for right-angled Artin groups. • A distance formula in the style of Masur–Minsky. The most recent results on the geometry of the action of the right-angled Artin group on its extension graph are contained in [4].

References [1] Brian Bowditch. Tight geodesics in the curve complex. Invent. Math. 171 (2008) 281–300. 62 10. GROUPS ACTING ON CUBICAL COMPLEXES AND SPACES WITH WALLS

[2] Sang-hyun Kim and Thomas Koberda. Embedability between right-angled Artin groups. Preprint. [3] Sang-hyun Kim and Thomas Koberda. An obstruction to embedding right- angled Artin groups in mapping class groups. To appear in Int. Math. Res. Notices. [4] Sang-hyun Kim and Thomas Koberda. The geometry fo the curve complex for a right-angled Artin group. In preparation. [5] Thomas Koberda. Right-angled Artin groups and a generalized isomorphism problem for ﬁnitely generated subgroups of mapping class groups. To appear in Geom. Funct. Anal. [6] Howard Masur and Yair Minsky. Geometry of the complex of curves. I. Hyper- bolicity. Invent. Math. 138, no. 1, 103–149, 1999. [7] Howard Masur and Yair Minsky. Geometry of the complex of curves. II. Hier- archical structure. Geom. Funct. Anal. 10, 902–974, 2000.

Sang Rae Lee My research area is Geometric Group Theory. In particular I am interested in CAT(0) cubical complexes and finiteness properties of groups. Recently I have constructed a family of finitely presented amenable groups with infinite virtual first Betti numbers. Those groups are first known examples which answer the following question in negative direction.

Qeustion 1. If a finitely presented group has infinite virtual first Betti number then must it be large? Does it contain F2?

Those group appear to be abelian-by-Houghton’s groups. The definition of Houghton’s groups are quite simple. For each positive inter n, we think of a discrete set Yn which is the disjoint union of n rays of discrete points emanating from the origin on the plane. A Hougthon’s group Hn is defined to be the group of eventual transla- tions on Yn which act on each ray as a translation outside a finite set. The action of Hn on the underlying set Yn defines a group AHn := (⊕Z) o Hn for each n ∈ N. Yn One interesting property of Houghton’s groups is that Hn has type of FPn−1 but not FPn (Brown). One way to see this is that Hn acts on n-dimensional CAT(0) cubical complexes satisfying some conditions. Currently I have been studying n- dimensional CAT(0) cubical complexes on which AHn acts to answer the question.

Question 2. Does AHn have the same ﬁniteness type of Hn?

I am also interested in CAT(0) cubical complexes where some ‘twisted subgroups’ of Houghton’s groups acts. By studying those complexes, I expect to answer the following question.

Question 3. Are there groups of type Fn (FPn) but not Fn+1 (FPn+1) which do not contain Z2 subgroup (n ≥ 3)? RUDOLF ZEIDLER 63

Tomasz Odrzygozdz See page 26.

Yulan Qing My research focuses on the boundaries of CAT(0) groups, specifically, Right-angled Artin groups and Right-angled Coxeter groups. I give an explicit and elementary proof that the Croke-Kleiner space does not have unique G-equivariant boundary π if we fix the gluing angle at 2 and changes the translate lengths of the generators of its fundamental group. This is a specific and elementary construction of a more general result studied by Croke and Kleiner in the context of geodesic flows in graph manifolds. I also prove that if a Right-angled Coxeter group acts geometrically on π this space, then the gluing angle has to be 2 . The immediate projects I’d like to pursue is to describe the Poisson boundary for the space and generate the second result to a larger class of graph of spaces. In general I am interested in questions about low-dimensional topology and geometric group theory centered around the boundaries of CAT(0) spaces and the groups which act geometrically on these spaces. The boundary of a CAT(0) space on which a given group acts can detect certain group structures that are difficult to see oth- erwise. While for many classes of groups there is no known type of boundary that is most natural for the group, different boundaries can be used to ”see” different structures about the group and the actions in question.

Emily Stark See page 42.

Rudolf Zeidler I am a master student at University of Vienna, currently in my fifth year of studying mathematics. I have been specializing my studies on geometry and topology and have become interested in geometric group theory. In fall of 2012, I have started working on my master thesis under the supervision of Prof. Goulnara Arzhantseva. The main goal of my thesis is to study coarse median structures on many examples of finitely generated groups. A coarse median space is a coarse generalization of the notion of a median algebra. It was introduced and studied by Bowditch [1]: A coarse median is a ternary operation µ : X × X × X → X on a geodesic space X which satisfies the axioms of a median algebra up to bounded error. (In a certain precise sense.) The existence of a coarse median is a quasi-isometry invariant of the underlying space and thus the theory is applicable to finitely generated groups. A metric space that admits a coarse median has several interesting properties: For example, one can bound the dimension of a quasi-isometrically embedded Eucledian space based on the “rank” of the coarse median. Moreover, a coarse median space satisfies at most quadratic isoperimetric inequality. Natural examples of coarse median spaces are Gromov hyperbolic spaces and CAT(0) cube complexes. In my master thesis, I first describe coarse medians on C0(1/6) small cancellation groups. – On the one hand, such a group is hyperbolic, and on the other hand, by Wise [2], it acts geometrically on a CAT(0) cube complex. Thus, there are two a priori different ways of defining a coarse median associated to such a group; I 64 10. GROUPS ACTING ON CUBICAL COMPLEXES AND SPACES WITH WALLS compare these two and determine how they are related. A further goal is to consider an infinite hyperbolic group with Kazdahn’s property (T), and describe a coarse median on it explicitly. In that case, unlike the previous examples, property (T) prevents the group from acting properly on a median space, cf. [3], but nevertheless it is coarse median due to hyperbolicity. By participating at this conference, I intend to broaden my knowledge in geometric group theory. Specifically, in view of the tutorial lectures, I am looking forward to learning more about groups acting on CAT(0) cube complexes, which is highly relevant to my master thesis. Also, it is a great opportunity to interact with researchers in this field and I hope to benefit from the knowledge of others as well as to contribute my ideas. References [1] B. H. Bowditch, Coarse median spaces and groups. Preprint (2011) [2] D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. [3] I. Chatterji, C. Drut¸u and F. Haglund, Kazhdan and Haagerup properties from the median viewpoint, Adv. Math. 225 (2010), no. 2, 882–921. CHAPTER 11

Bounded cohomology, simplicial volume and quasimorphisms

Matthias Blank We are interested in so called bounded cohomology. In order to construct bounded cohomology, instead of looking at general cochains as in regular cohomology of topological spaces, one only considers bounded cochains (with respect to the canonical `1-norm on the chain complex). This leads to a theory very different from ordinary cohomology. First one notes, that bounded cohomology encompasses the so called simplicial volume of a closed oriented manifold, which in important cases corresponds to the Riemannian volume of the manifold. On the other hand, bounded cohomology essentially depends only on the fundamental group of the corresponding space and can thus be viewed as a pure group theoretical tool. Hence bounded cohomology provides a link between (Riemannian) geometry, topology and group theory. This is exemplified in the seminal role it plays in Gromov’s proof of Mostow’s Rigitity Theorem. In the calculations of bounded cohomology, ideas from geometric group theory are often central. Conversely, bounded cohomology can for instance detect whether a group is hyperbolic (or amenable) or not. The theory is both enriched and complicated by the lack of accessible tools (like excision) for more general calculations. A fundamental exception to this is the existence of a relative version of bounded cohomology, essentially due to Gromov. Though there exists now a variety of constructions of relative bounded cohomology, there seems to be no optimal one. We hope to be able to design a new one that could be at least complementary to the existing ones. For example, we hope to find a homological algebraic description of resolutions calculating relative bounded cohomology and the correct norm. The script by C. Löh[2] is a great starting point for getting acquainted with the subject. It also contains a comprehensive bibliography. The pivotal point of the subject remains the fundamental work by M. Gromov [1]. References [1] M. Gromov, Volume and bounded Cohomology, Publ. Math. IHES, 56 (1982). [2] C. Löh, Group Cohomology and Bounded Cohomology. An Introduction for Topologists, Lecture Notes (2010).1

1Available online via www.mathematik.uni-regensburg.de/loeh/teaching/topologie3 ws0910/prelim.pdf.

65 66 11. BOUNDED COHOMOLOGY

Claire Burrin With (too much) delay, I heard of the first YGGT meeting and discovered via the website its program with much appetite. I recently finished my Master thesis at ETH Zurich and am starting now my PhD studies there under the continued supervision of Prof. Alessandra Iozzi. My Master thesis centered on a recent article of N. Ozawa [2]. Following the direction taken by Burger and Monod’s refinement of property (T) in the context of bounded cohomology – the so-called property (TT) (see [1]) – the author presents a further strengthening, aptly named property (TTT). The motivation behind property (TTT) is to introduce more rigidity in the in- compatibility between locally compact groups with property (T) (resp. (TT)) and a-T-menable groups (resp. a-T-menable and hyperbolic groups). It can be established from the definitions of these properties that any continuous homomorphism from a group of the first type to a group of the second type is trivial in the sense that it has compact image. If we are to consider more broadly quasi-homomorphisms, then the statement is made valid under the condition that the domain group is a property (TTT) group. In particular, the higher-rank algebraic groups SLn(K) (for n ≥ 3 and K any local field) and their lattices are shown to possess this property. The purpose of my work was to expose in its full context this new property and render the author’s work in the most accessible manner I could. There are some open questions to be studied, such as a more precise understanding of the relations between property (TTT) and variations around it (as introduced in [2]) that might prove to be equivalent, or yet still, the exploration of a parallel concept of a-TTT-menability and the search for examples of groups with this property other than a-T-menable and hyperbolic groups. My research might involve a more extensive look into these questions or concentrate on a different subject of geometric group theory, for which the participation to this YGGT meeting seems an excellent opportunity. References [1] N. Monod, ”Continuous bounded cohomology of locally compact groups”. Lect. Notes in Math. 1618 (2001), Springer. [2] N. Ozawa, Quasi-homomorphism rigidity with noncommutative targets. J. R. Angew. Math. 655 (2011), 89-104.

Caterina Campagnolo After my master thesis on growth functions of right-angled Coxeter groups, I began in May 2011 a PhD under the supervision of Michelle Bucher. My aim is to study characteristic classes of surface bundles, as deﬁned by Morita (see [2]). In particular, Morita asked whether these classes are bounded, or in other words, if they can be represented by cocycles which are uniformly bounded. This is known to hold for the classes in degree 2(k +1) since these are pullbacks of primary classes on the symplectic group, which are bounded by a result of Gromov. The question for the remaining classes in degree 2k is open from degree 4 already. One advantage of the theory of bounded cohomology, initiated by Gromov in the beginning of the 80’s [1], is that good bounds for norms of cohomology classes naturally give rise to Milnor-Wood inequalities. I will thus try to compute the norms of the characteristic classes of surface bundles, with as aim to produce new inequalities between classical invariants of surface bundles. CRISTINA PAGLIANTINI 67

I spent the first year of my thesis learning mainly about general and bounded cohomology of groups and about the mapping class group of surfaces. In fact, characteristic classes of surface bundles are, in the universal case, cohomology classes of the mapping class group At the moment, I am studying various classes in the second cohomology group of the mapping class group, in particular Meyer’s signature cocycle (see [3]), and I am trying to find representatives with low norm. References [1] M. Gromov, Volume and bounded cohomology, Inst. Hautes Etudes´ Sci. Publ. Math. No. 56, (1982), 5–99 (1983). [2] S. Morita, The Geometry of Characteristic Classes, American Mathematical Society, 2001. [3] W. Meyer, Die Signatur von Flächenbündeln. Math. Ann. 201, pp. 239-264, 1973.

Dmitry Kagan A real-valued function f on a group G is called a homogeneous quasimorphisms if the set {f(xy) − f(x) − f(y); x, y ∈ F } is bounded and f(xn) = nf(x) ∀n ∈ Z, ∀x ∈ F . R.I. Grigorchuk raised the following question about the existence of non- trivial homogeneous quasimorphisms on groups with one deﬁning relation, and two generators:

Let G be a non-amenable group with one deﬁning relation. Is it true that there are (2) non-trivial homogeneous quasimorphisms on G and that Hb (G) 6= 0?

We have answered this question in the case where the group G has non-trivial center: Theorem. Let G be a group with one defining relation and nontrivial center. Then on G there exist nontrivial homogeneous quasimorphism, with the exception of the following cases: The group G is cyclic, free abelian G = ht, a|ta = ati or G is metabelian, i.e. it has a representation of the form G = ht, a|tat−1 = api or G = ht, a|t2 = a2i. Existence of non-trivial homogeneous quasimorphisms implies in particular that (2) Hb (G) 6= 0 and that the verbal subgroup V (G) has infinite width relative to a finite proper set of words V from a derived subgroup.

Cristina Pagliantini I am currently interested in the estimation of simplicial volume of manifolds with non-empty boundary and its behaviour under topological operations (see [1,2]). Gromov introduced the simplicial volume in his pioneering work Volume and bounded cohomology [4], published in 1982. The simplicial volume is a homotopy invariant of compact manifolds defined via a natural `1-seminorm on real singular homology. More precisely, for an oriented manifold it is the seminorm of the real fundamental class of the manifold itself. As a powerful tool for computing the simplicial volume, Gromov himself developed the theory of bounded cohomology [4]. The complex of bounded cochains is defined as the subcomplex of the singular cochains which are bounded with respect to the 68 11. BOUNDED COHOMOLOGY supremum norm, and it coincides with the topological dual of the complex of singular chains endowed with the `1-norm, mentioned above. Bounded cohomology is the cohomology of this subcomplex and it turns out to be a fundamental instrument in order to estimate `1-norms, mainly via the study of the usual Kronecker pairing between homology and cohomology. In order to study simplicial volume in the relative setting of manifolds with nonempty boundary, I would like to develop some aspects of the (continuous) bounded cohomology of pairs of spaces and groups extending Gromov’s definition (see [3]). Furthermore bounded cohomology constitutes a fascinating theory by itself since allows to establish a link between topology and group theory. References [1] M. Bucher, R. Frigerio, C. Pagliantini The simplicial volume of 3-manifolds with boundary, arXiv:1208.0545. [2] R. Frigerio, C. Pagliantini, The simplicial volume of hyperbolic manifolds with geodesic boundary, Algebr. Geom. Topol. 10, pp. 979-1001, 2010. [3] R. Frigerio, C. Pagliantini, Relative measure homology and continuous bounded cohomology of topological pairs, Pacific J. Math. 257, pp. 91-130, 2012. [4] M. Gromov. Volume and bounded cohomology, Inst. Hautes Etudes´ Sci. Publ. Math. 56, pp. 5–99, 1982.

Hester Pieters After ﬁnishing my master thesis on hyperbolic Coxeter groups and the Leech lattice at the Radboud University Nijmegen, I began in September 2011 a PhD under the supervision of Michelle Bucher at the University of Geneva.

I am interested in the connection between the topology and the geometry of manifolds. On Riemannian manifolds, typically of nonpositive curvature, invariants from topology such as for example characteristic numbers are often proportional to geometric invariants such as the volume. An example are the so called Milnor-Wood inequalities which relate the Euler class of ﬂat bundles to the Euler characteristic of the base manifold.

More speciﬁcally, I use techniques from continuous bounded cohomology to study such questions. Lately I have been interested in the natural comparison map that exists between continuous bounded cohomology and usual continuous cohomology. I have been looking at the proof of the injectivity of the comparison map in degree 3 for the isometries of the 3-dimensional real hyperbolic space which follows from a result of Bloch and I intend to generalize this.

I am furthermore interested in cohomology classes of complex hyperbolic manifolds. Let M be a manifold and let β ∈ Hq(M; R) be a cohomology class. The Gromov norm kβk∞ is by definition the infimum of the sup-norms of all cocycles representing β: kβk∞ = inf{kbk∞ | [b] = β} ∈ R≥0 ∪ {+∞} This is a way of assigning a numerical invariant to a cohomological invariant. It has for instance been computed for the Kählerclass. However, the value of this norm is only known for a few cohomology classes. I would like to determine its value for certain cohomology classes of complex hyperbolic manifolds, especially in top PASCAL ROLLI 69 dimension. This could lead to new Milnor-Wood type inequalities and computations of the simplicial volume of complex hyperbolic manifolds.

Beatrice Pozzetti I am a second year PhD student at ETH and I am mainly interested in the study of bounded cohomology (both of groups and of topological spaces) and in its applications to geometry and representation theory. Bounded cohomology is an exotic cohomological theory defined by Gromov in [4] that has many relations with hyperbolicity and rigidity but, since it is extremely hard to compute, it is still poorly understood. Apart from a general attempt of understanding better the theory in itself [1], my research focuses on the applications of bounded cohomology to the study of representations of discrete groups into Lie groups of Hermitian type. In this field bounded cohomology can be used to define numerical invariants that allow us to select some preferred components of the representation variety [3]. If, for example, we consider representations of surface groups in PSL2(R), the components selected by this bounded cohomological invariant correspond to the Teichmuller space. In [2] Burger and Iozzi showed a remarkable rigidity result for representations of lattices in PU(n, 1) with target the group PU(m, 1) that maximize the invariant defined from bounded cohomology: they are equivariant with respect to a totally geodesic holomorphic embedding of Hn in Hm. I’m trying to generalize this result C C to representations with target any Lie group of Hermitian type. References [1] M. Bucher, M. Burger, R. Frigerio, A. Iozzi, C. Pagliantini, M. B. Pozzetti. Isometric properties of relative bounded cohomology. Preprint, 2012. [2] M. Burger, A. Iozzi. A measurable Cartan theorem and applications to deformation rigidity in complex hyperbolic geometry. Pure Appl. Math. Q. 4, 2008. [3] M. Burger, A. Iozzi, A. Wienhard. Surface group representations with maximal Toledo invariant. Ann. Math., 172, 2010. [4] M. Gromov. Volume and bounded cohomology. Publ. Math. IHES 56 (1982) 5–99.

Pascal Rolli The topics of my research are quasimorphisms and bounded cohomology. I am currently working on generalizations of my own combinatorial construction of a class of ∞ quasimorphisms qσ : F → R, σ ∈ ` , defined on free groups F. This construction is rather different from previously known quasimorphisms, in particular it is linear and leads to a rather simple proof of the infinite-dimensionality of the second bounded cohomology of non-abelian free groups (see http://arxiv.org/abs/0911.4234). Firstly, I recently reinterpreted the construction using actions on trees. This led to a uniform description of a class of quasimorphisms assigned to an action of a group Γ on a tree T . It applies in particular to amalgams Γ = A ∗B C and HNN- extensions Γ = A∗ϕ acting on their Bass-Serre trees. I am currently studying these situations and I have obtained certain non-triviality results. The relevant bounded cohomology spaces have been investigated by K. Fujiwara who proved infinite- dimensionality using a generalization of a construction by R. Brooks. Again, it is my aim is to utilize the linearity of my construction to obtain infinite-dimensional subspaces of bounded cohomology. 70 11. BOUNDED COHOMOLOGY

Secondly, I’ve been studying twisted 2-cocycles for linear isometric actions of free products Γ = A ∗ B on Hilbert spaces. These are maps that reduce to quasimorphisms when the action is trivial. The construction given in the aforementioned article has a natural generalization to this setting. I was able to prove that every action of Γ on a finite-dimensional Hilbert space H admits non-trivial twisted 2- cocycles. In the case of H being infinite dimensional, most importantly if H = `2(Γ), the arguments are no longer valid. I’m currently trying to fix this by using the CAT(0)-structure of the metric in H. Furthermore, I constructed a quasimorphism in a purely geometric way, using the action of a free group on a certain CAT(0)-square complex quasi-isometric to the group’s Cayley graph.

Karol Strzakowski I have just started my PhD studies at the Polish Academy of Sciences. My scientific interests are in general algebraic topology, in particular its applications to geometry, and geometric group theory. I am currently looking for an adequate topic for my PhD thesis, at the moment there are two possibilities I am interested in. The first one, related to widely understood geometric group theory, is studying simplicial volume, defined by Gromov [1] for compact manifold M as:

kMk = inf{kck1 | c is a cycle representing fundamental class of M} In many cases computations are made using bounded cohomology. I am especially interested in studying generalization of this invariant for non-compact manifolds, obtained by using locally finite homology, and product formulas. It is well known that there are following inequalities for simplicial volume of a product M × N: n + m kMkkNk ≤ kM × Nk ≤ kMkkNk n where m = dimM, n = dimN. These inequalities are in general almost never sharp and in many special cases one can prove better estimations. However, there are only a few cases where preceise formulas are known. The second one, more related to algebraic topology and geometry, is working on Chern-Schwarz-MacPherson classes, which are generalizations of classical Chern classes, suitable to work with singular algebraic varieties. The existence of such classes was conjectured by Deligne and Grothendieck and they were eventually constructed independently by Marie-HélèneSchwartz [2] and Robert MacPherson [3]. I am especially interested in conjecture given by Paolo Aluffi and Constantin Mihalcea [4], who were studying Chern classes of Schubert cells, i.e. cells present in the cell decomposition of a complex Grassmannian. The total Chern classes of such cells can be obviously represented as a linear combination of fundamental classes of Schubert varieties (which are the sums of suitable cells) with integer coefficients. The conjecture states that all these coefficients are in fact positive, what seems to have some, but also mysterious, combinatorial consequences. References [1] M. Gromov Volume and bounded cohomology, Institut des Hautes Etudes´ Scien- tifiques Publications Mathématiques No. 56 (1982), 5-99. KAROL STRZAKOWSKI 71

[2] M.-H. Schwartz Classes charactéristiquesdéfinespar une stratification d’une variétéanalytique complexe I, II, Comptes Rendus de l’Académiedes Sciences, Paris, No. 260 (1965), 3262-3264,3535-3264. [3] R. D. MacPherson Chern classes for singular algebraic varieties, Annals of Math- ematics, No. 100 (1974), 423-432. [4] P. Aluffi, C. Mihalcea Chern classes of Schubert cells and varieties, arXiv:math/0607752, 2006.

CHAPTER 12

Locally-compact groups, diﬀeomorphism groups, other non-discrete groups

Mathieu Carette My research is currently focused on locally compact hyperbolic groups and convergence groups. The definition of a locally compact group is hyperbolic extends naturally that of discrete hyperbolic groups : we ask that the group is compactly generated and word-hyperbolic with respect to a compact generating set. Genuinely new phenomena occur while broadening the setting to locally compact groups : there exist interesting amenable hyperbolic groups, as well as hyperbolic groups acting transitively on their (uncountable) boundary. Recent results by Caprace, de Cornulier, Monod and Tessera [2] show that one can describe the structure of these hyperbolic groups very precisely. A celebrated result of Bowditch [1] shows that one can characterize discrete hyperbolic groups acting on their boundary dynamically/topologically. In recent joint work with Dennis Dreesen, I have generalized this characterization for locally compact groups. This yields insight on natural questions about locally compact groups acting continuously on compact sets. In particular, one can hope to classify locally compact groups G acting continuously and sharply 3-transitively on a compact set X. When X is connected a general result due to Tits [3], [4] implies in particular that G is the real or complex projective group P GL2 acting on the projective line. Projective groups over non-Archimedean local fields (e.g. Qp) yield examples of locally compact groups acting properly and cocompactly on locally finite trees such that the action on the (compact, totally disconnected) boundary is sharply 3-transitive. I am interested in studying these groups, and determining whether these are the only examples. Our work also leads to a characterization of locally compact transitive convergence groups in the spirit of [2]. I am currently working on possible applications of this characterization to rank 2 topological buildings, namely topological generalized polygons. References [1] B.H. Bowditch, A topological characterization of hyperbolic groups, J. Amer. Math. Soc. 11, No. 3 (1998), 643–667. [2] P.E. Caprace, Y. de Cornulier, N. Monod, R. Tessera, Amenable hyperbolic groups, preprint. [3] L. Kramer, Two-transitive Lie groups, J. Reine Angew. Math. 563 (2003), 83–113. [4] J. Tits, Sur les groupes doublement transitifs continus, Comm. Math. Helv. 26 (1952), 203–224.

73 74 12. NON-DISCRETE GROUPS

Corina Ciobotaru See page 22.

Amichai Eisenmann See page 47.

Lucas Fresse My research is in the domain of geometric representation theory. My main objects of study are the nilpotent orbits in reductive Lie algebras and the geometric objects that can be related to them. To a nilpotent element in a reductive Lie algebra, one can associate several algebraic varieties which play roles in geometric representation theory: examples are Springer ﬁbers and orbital varieties. In general, the geometric properties of these objects are not well understood. In recent works, I have studied questions on the geometry of Springer ﬁbers and orbital varieties: existence of cell decompositions, singularity of their irreducible components, actions of groups, decomposition into orbits, property to be spherical. This topic has algebraic, geometric and combinatorial aspects.

Swiatoslaw Gal Classical Geometric Group Theory, founded inter alia on the Milnor–Svarcˇ lemma, studies cocompact proper isometric actions and variations thereof (cf. Farrel–Jones conjecture). Studying topological or metric properties of such spaces we can deduce nontrivial group properties. Theorem (Haglund). Assume that a finitely generated group Γ acts properly by isometries on a cubical CAT(0) complex. Then every element of Γ is undistorted. Surprisingly one can obtain the very same conclusion under completely different assumptions. Theorem (Polterovich). Assume that a finitely generated group Γ acts by Hamil- tonian diffeomorphisms on a compact symplectic manifold. Then every element of Γ is undistorted. Recently I am interested in searching for other obstructions for actions of discrete groups on topological spaces., as well as for constructions of such actions. A, possibly not so different, topic emanating from classical geometric group theory is about biinvariant metrics on groups. My favourite problem is the following. Question. Does there exist a cocompact lattice Γ in a simple Lie group of higher rank supporting an unbounded bi-invariant metric?

Lukasz Garncarek I am a second year PhD student at the Mathematical Institute of the Wroclaw University. My scientiﬁc interests revolve around group theory and geometry. So far, my research concerned analytic aspects of groups. Currently, I am working simultaneously on a few related problems, continuing the work from my MSc thesis, hoping that one of them will turn out to be a suitable topic for my PhD thesis. THOMAS HAETTEL 75

In my MSc thesis I studied the irreducibility of some representations of subgroups of the group Diffc(M) of compactly supported diffeomorphisms of a smooth manifold M. If we take a measure µ on M, having smooth density with respect to the Lebesgue measure in maps, we may define a unitary representation π of Diffc(M) on L2(M, µ) by dφ µ1/2+is (12.1) π (φ)f = ∗ f ◦ φ−1, s dµ where s ∈ R. The case of the group of compactly supported diffeomorphisms preserving a measure was described by Vershik, Gelfand and Graev. In my thesis I considered the irreducibility of representations (12.1) for the groups Symplc(M) and Contc(M) of compactly supported symplectomorphisms and contactomorphisms. After the “large” groups of diffeomorphisms, I studied representations of Thomp- son’s groups F and T , which are finitely presented, hence “small” in the combinatorial sense. Their natural actions on the unit interval and unit circle however still resemble the action of a “large” group, and the representations (12.1) of F and T turn out to be irreducible. Moreover, representations πs and πt are inequivalent, provided that s − t is not a multiple of 2π/ log 2. In the representation theory of SL2(R), the representations of the form (12.1), 2 associated to the natural action of SL2(R) on P(R ), form a part of the principal series. They are induced from one-dimensional representations of subgroups of SL2(R). This is not the case for the Thompson’s groups. In fact, πs are nonequiv- alent to representations induced from finite-dimensional representations of proper subgroups of F or T . Hence, the two possible generalizations of the principal series to F and T are disjoint. One of the problems I am working on is to generalize the above results for quasi- invariant actions of discrete groups on arbitrary measure spaces. In other words, I want to understand how the irreducibility of the representations πs is related to dynamical properties of the action. My work on representations of the group of contactomorphisms has inspired the following question: Problem. Let G be a topological group. Suppose that G contains no nontrivial compact subgroups. Does it imply that the convolution algebra Mc(G) of compactly supported complex Borel measures on G has no zero divisors? The positive answer in the case of Rn is a variant of the Titchmarsh convolution theorem. The answer for locally compact abelian groups follows from the work of Benjamin Weiss. In a recent preprint I managed to give a positive answer for supersolvable Lie groups. I want to continue this work and extend my results to a wider class of groups.

Thomas Haettel See page 52.

CHAPTER 13

Other aspects of geometric group theory: Algebraic geometry, Thompsons groups, tree manifolds

Abhishek Banerjee The objective of my research is to systematically develop the Algebraic Geometry of schemes over a symmetric monoidal category (C, ⊗, 1). Our notion of a scheme over the category C is that of To¨enand Vaqui´e. In this context, I have worked with Picard groups as well as the derived algebraic geometry (involving simplicial monoids) over the category C. When C = k − Mod, the category of modules over a commutative ring k, this reduces to the usual algebraic geometry of schemes over Spec(k).

My current research focus in this direction is that on group schemes over a symmetric monoidal category (C, ⊗, 1). It is natural to ask whether, as with algebraic geometry, the arithmetic geometry can be similarly developed in the general framework of symmetric monoidal categories. In particular, since the theory of finite flat group schemes is closely linked to arithmetic, they are a natural starting point for such a theory. In this context, I proved an analogue of Deligne’s lemma on the order of elements in affine, flat, commutative group schemes of finite rank. The methods described above can be applied, more generally, to groupoid schemes over a symmetric monoidal category. In this direction, I hope to prove results relating quasi-coherent sheaves on groupoid schemes to quasi-coherent sheaves on simplicial schemes. These results will enhance the understanding of notions of descent over symmetric monoidal categories.

Ariadna Fossas My scientific interests concern Geometric Group Theory, Combinatorial Group The- ory and Topology, and I am mainly working on R.J. Thompson’s groups of type T . Thompson’s group T is usually seen as a subgroup of the group of dyadic, piecewise linear, orientation-preserving homeomorphisms of the unit circle. However, T can also be identified to a group of equivalence classes of balanced pairs of binary trees, to a subgroup of piecewise PSL(2, Z), orientation-preserving homeomorphisms of the projective real line, and to the asymptotic mapping class group of a fattened complete trivalent. In my PhD thesis, I first prove that the canonical copy of PSL(2, Z) obtained from “piecewise PSL(2, Z)” is a non-distorted subgroup of T . As a corollary, T has non-distorted subgroups isomorphic to F2. The second result of the thesis uses “modular T ” to state that there are exactly ϕ(n) (Euler function) conjugacy classes

77 78 13. OTHER ASPECTS OF GEOMETRIC GROUP THEORY of elements of order n. The third result constructs a minimal simply-connected (conjectured contractible) cellular complex C on which the group T acts by automorphisms. The automorphism group of C is “essentially” T itself. The complex C can be seen as an infinite generalization of Stasheff’s associahedra. On the final part of the thesis a geometric boundary at infinity of C1 is defined. Future research prospects follow three main axes. The first one is concerned with the geometric compactification of C1 and the possible applications to the study of analytic properties of T . The second axis concerns the generalized Thompson groups. The three classical groups, F , T et V have been generalized by Higman, Brown, Stein, River and others. Moreover, Greenberg and Sergiescu, Brin, and Dehornoy defined braided versions of those generalizations. It is thus natural to ask which of the definitions and properties introduced in my thesis have good analogues in generalized Thompson groups and their braided versions. The third and last axis concerns the “modular” point of view on T , due to Funar-Kapoudjian-Sergiescu, which relates T to modular groups. The result on the automorphisms group of C can thus be interpreted as an analogue of Ivanov’s result describing modular groups as automorphisms groups of curve complexes. It could be interesting to study the resemblances and differences between T and the classical modular groups. Even though I have only mentioned questions related to my PhD thesis, I am interested in other areas of Geometric Group Theory and connected domains.

Pawe lZawi´slak My current investigations are focused on the properties of trees of manifolds with boundaries and boundaried trees of manifolds. Trees of manifolds are inverse limits of certain inverse systems of compact manifolds. The most familiar examples of such spaces are the Pontryagin spheres, orientable and nonorientable, which are trees of 2-tori and of projective planes, respectively. The first such space was constructed by W. Jakobsche (see [5]) as a potential counterexample to the Bing-Borsuk conjecture. Ealier similarly constructed spaces were considered in different context by L.S. Pontryagin (see [9]) and R.F. Williams (see [12]). Then F.D. Ancel and L.C. Siebenmann (see [1]) noticed that the tree of some homological 3-spheres can be identified with a compactification of the Davis contractible 4-manifold which covers a closed 4-manifold (see [3]). Properties of trees of orientable manifolds were investigated later by W. Jakobsche. In [6] he showed m-homogenity of these spaces (for every natural m). Moreover, he showed that a tree of orientable homology spheres is a cohomology manifold, the space which can often be identified with the fixed-point set of a topological action on a manifold or a cohomology manifold. He showed also that such homogeneous cohomology manifolds appear as compactifications of contractible 4-manifolds, or orbit spaces of actions of 0-dimensional infinite compact groups. Some results of W. Jakobsche were extended to the nonorientable case by P.R. Stallings (see [11]). In Jakobsche’s original definition a tree of manifolds is the inverse limit lim (L , α ) ←− n n of a certain inverse systems of orientable manifolds Ln and maps αn : Ln → Ln−1 between them (see [6]). This inverse system has the property that every element Ln+1 is, up to a homeomorphism, a connected sum of its predecessor Ln and a

finite number of closed, orientable manifolds Ln,1,...,Ln,kn , which belong to a countable family L. Jakobsche showed that if the inverse system (Ln, αn) satisfies PAWEL ZAWISLAK´ 79 some natural assumptions, then the inverse limit lim (L , α ) depends only on the ←− n n family L. We denote this inverse limit by X (L). Later P.R. Stallings generalized this definition to the case of nonorientable PL- manifolds (see [11]). This suggests the question about further generalisations of the above definitions. A first natural generalisation is provided by the notion of a boundaried tree of manifolds. These spaces are the inverse limits lim (L , α ) of certain inverse systems ←− n n of compact manifolds Ln with boundaries and maps αn : Ln → Ln−1 between them. Every element Ln+1 of such system arise from its predecessor by performing finitely many operations of one of the following types: a connected sum with a compact manifold (with or without boundary) belonging to a countable family L0, or a boundaried sum with a compact manifold with boundary belonging to a countable family L1. At the moment the following properties of boundaried trees of manifolds are known:

• I showed (see [14]) that if the inverse system (Ln, αn) satisfies some natural assumption generalizing Jakobsche’s assumptions, then the inverse limit lim (L , α ) depends only on the families L0 and L1 (we denote this inverse ←− n n limit by X (L0, L1)), • J. Swi¸atkowski´ showed (private conversation) that the topological dimen- 0 1 sion of the space X (L , L ) is equal to the dimension of L1 minus 1 (provided that the family L1 is not empty). The following properties of boundaried trees of manifolds are believed to hold: • no local cut-points, • the group of homeomorphisms of X (L0, L1) acts transitively on points coming from the interior of Ln and from the connected components of the boundary of Ln. D. Osajda showed in [8] that the Gromov boundary of a 7-systolic simplicial complex X (see [7] for the definition) is homeomorphic to the inverse limit lim (S , Π ) of ←− n n a system of combinatorial spheres Sn centered at a fixed vertex v ∈ X and natural projections Πn : Sn → Sn−1 between them. In the case when the complex X is a normal pseudomanifold of dimension 3, it appears that these spheres are surfaces and the Gromov boundary ∂GX is a tree of surfaces (orientable or not, according to the orientability or nonorientability of X) (see [13]). H. Fischer showed in [4] that trees of manifolds appears naturally as boundaries of right-angled Coxeter groups with manifold nerves. More precisely, if the nerve of a right-angled Coxeter group is PL-homeomorphic to the triangulation of a closed, oriented manifold M, then the CAT (0) boundary of this group is homeomorphic to the tree X ({M, M¯ }), where (M¯ is M with the opposite orientation) In [10] P. Przytycki and J. Swi¸atkowski´ showed that every closed 3-manifold M admits a ”flag-no-square” triangulation. It follows that there exist hyperbolic right- angled Coxeter groups, which nerves are PL-homeomorphic to M, and thus which Gromov boundaries are homeomorphic to X ({M, M¯ }). J. Swi¸atkowski´ showed (private conversation) that if X is a CAT (0) complete, piecewise euclidean or piecewise hyperbolic, oriented simplicial pseudomanifold of dimension n + 1, which singularity locus has dimension 0 (i.e. links of points are n-spheres, except the vertices, where triangulations of closed n-manifolds from a family M can appear), then the CAT (0)-boundary of X is homeomorphic to X (M). 80 13. OTHER ASPECTS OF GEOMETRIC GROUP THEORY

Such pseudomanifolds exist due to Charney-Davis hyperbolisation procedure (see [2]) Thus the natural question appears, if the boundaried trees of manifolds appear as boundaries of nonpositively curved spaces and groups. References [1] F.D. Ancel and L.C. Siebenmann, The construction of homogenous homology manifolds, Abstracts Amer. Math. Soc. 6 (1985), 92. [2] R.M. Charney i M.W. Davis, Strict hyperbolization, Topology 34 (1995), no.2, 329-350. [3] M.W. Davis, Groups generated by reflecions and aspherical manifolds not covered by euclidean space, Ann. of Math. [4] H. Fischer, Boundaries of right-angled Coxeter groups with manifold nerves, Topology 42 (2003), 423-446. [5] W. Jakobsche, The Bing-Borsuk conjecture is stronger than the Poincare conjecture Fund. Math. 106 (1980), 127-134. [6] W. Jakobsche, Homogenous cohomology manifolds which are inverse limits, Fund. Math. 137 (1991), 81-95. [7] T. Januszkiewicz and J. Swi¸atkowski,´ Simplicial nonpositive curvature, Publ. Math. IHES 104 (2006), no.1, 1-85. [8] D. Osajda, Ideal boundary of 7-systolic complexes and groups, Algebraic & Geo- metric Topology 8 (2008), 81-99. [9] L.S. Pontryagin, Sur une hypothesésefundamentale de la théoriede la dimension, C. R. Acad. Sci. Paris 190 (1930), 1105-1107. [10] P. Przytycki and J. Swi¸atkowski,´ Flag-no-square triangulations and Gromov boundaries in dimension 3, Groups Geom. Dyn. 3 (2009), 453-468. [11] P.R. Stallings An extension of Jakobsche’s construction of n-homogenous continua to the nonorientable case, Continua (with the Houston Problem Book), ed. H. Cook, W.T. Ingram, K. Kupenberg, A. Lelek, P. Minc, Lect. Notes in Pure and Appl. Math. vol. 170 (1995), 347-361. [12] R.F. Williams, A useful functor and three famous examples in topology. Trans. Amer. Math. Soc. 106 (1963), 319-329. [13] P. Zawi´slak, Trees of manifolds and boundaries of systolic groups, Fund. Math. 207 (2010),71-99. [14] P. Zawi´slak, Boundaried trees of manifolds, in preparation. Alphabetical List of Contributors

Antolin Pichel, Yago, 58 Koone, Shelley, 44 Arnt, Sylvain, 59 Kwon, Sanghoon, 48

Banerjee, Abhishek, 77 Le Boudec, Adrien, 6 Blank, Matthias, 65 Lee, Sang Jin, 53 Burrin, Claire, 66 Lee, Sang Rae, 62 Byron, Ayala, 1 Maloni, Sara, 41 Campagnolo, Caterina, 67 Marcinkowski, Michal, 15 Carette, Mathieu, 73 Martin, Alexandre, 32 Chao, Khek Lun Harold, 5 Matte Bon, Nicolas, 25 Ciobotaru, Corina, 22, 74 Matucci, Francesco, 8 Clais, Antoine, 51 Myropolska, Aglaia, 2 Cohen, Moshe, 44 Nguyen, Thang, 8 Degrijse, Dieter, 15 Nicolae, Adriana, 26 Der´e,Jonas, 23 Niesdroy, Henning, 49 Diana, Francesca, 14 Odrzygozdz, Tomasz, 26, 63 Disarlo, Valentina, 37 Durham, Matthew, 37 Pagliantini, Cristina, 68 Duszenko, Kamil, 52 Parzanchevski, Ori, 2 Pieters, Hester, 69 Eisenmann, Amichai, 47, 74 Pozzetti, Beatrice, 69 Fossas, Ariadna, 78 Przytycki, Piotr, 45 Fresse, Lucas, 74 Puder, Doron, 3 Qing, Yulan, 63 Gal, Swiatoslaw, 48, 74 Garncarek, Lukasz, 75 R¨oer,Malte, 17 Gruber, Dominik, 30 Raimbault, Jean, 16 Gupta, Radhika, 59 Rolli, Pascal, 70

Haettel, Thomas, 52, 75 Schwandt, Marco, 18 Hagen, Mark, 60 Schwer, Petra, 34, 53 Horbez, Camille, 38 Sisto, Alessandro, 26, 33 Hume, David, 31 Sklinos, Rizos, 4 Solomon, Yaar, 9 Judge, Chris, 39, 48 Stark, Emily, 42, 63 Kagan, Dmitry, 67 Strzalkowski, Karol, 71 Kangaslampi, Riikka, 24, 52 Swiecicki, Krzysztof, 35 Kent, Curtis, 7 Talambutsa, Alexey, 10 Kielak, Dawid, 39 Thumann, Werner, 27 Kim, Sanghyun, 61 Tonic, Vera, 19, 42 Koberda, Thomas, 40, 62 Koivisti, Juhani, 24 Wauters, Sarah, 20

81 82 ALPHABETICAL LIST OF CONTRIBUTORS

Weil, Steﬀen, 28 Whittle, Carrie, 4

Zadnik, Gasper, 11 Zawislak, Pawel, 80 Zeidler, Rudolf, 64