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

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Research Statements for the Second Young Geometric Group Theory Meeting at the Center for Mathematical Sciences, Technion, Haifa, Israel, February 2013 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 ar- eas. 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 re- search 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 geo- metric 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, fΓn(Gk)gn≥rank(Gk) is a family of expanders for a family of finite abelian groups fGkg [5]. Gambord and Pak [3] proved that the statement is true for the fam- ily fPSL(2; p)gp!1 [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 neces- sary, 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 prop- erty (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 2 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 2 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.
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