Geometric and Asymptotic Group Theory with Applications Geometric and Asymptotic Group Theory with Applications AMS Subject Classiﬁcation: 20-XX A

Geometric and Asymptotic Group Theory with Applications Geometric and Asymptotic Group Theory with Applications AMS Subject Classiﬁcation: 20-XX A. Martino, V. Shpilrain i E. Ventura (editors) Escola Polit`ecnica Superior d’Enginyeria de Manresa ISBN-13: 978-84-86784-06-5 ISBN-10: 84-86784-06-9

Disseny de portada: Estudi Mestres Composici´oen LATEX: A. Roig Impressi´o:Remsa S.L. Manresa Manresa, 31 de juliol de 2006 Geometric and Asymptotic Group Theory with Applications

Manresa (Catalunya) August 31st to September 4th, 2006

Organising committee: E. Ventura (coordinator) (Universitat Politècnica de Catalunya), V. Shpilrain (City College of New York), A. Martino (Universitat Politècnica de Catalunya), M. Sapir (Vanderbilt University), R. Gilman (Stevens Institute), J. González-Meneses (Universidad de Sevilla).

Scientific committee: E. Ventura (Universitat Politècnica de Catalunya), V. Shpilrain (City College of New York), A. Miasnikov (McGill Univerity), E. Zelmanov (University of California), C. Mart´ınez (Universidad de Oviedo), J. Burillo (Universitat Politècnica de Catalunya), J. Villar (Universitat Politècnica de Catalunya).

Local committee: M. Alsina (Universitat Politècnica de Catalunya), R. Caparrós(Universitat Politècnica de Catalunya), A. Roig (Universitat Politècnica de Catalunya). With the support of

Oﬁcina de Suport a la Recerca Matem`atica

Societat Catalana de Matem`atiques 1 Foreword

Wellcome everybody to the GAGTA conference, “Geometric and Asymptotic Group Theory with Applications”, held in Manresa (Catalonia) from August 31st to Septem- ber 4th, 2006.

This meeting is one of the 64 satellite activities of the International Congress of Mathematicians, ICM2006, that has been recently held in Madrid. On the occasion of the ICM2006 and after the success of the “Barcelona Group Theory Conference” held in July 2005 at the “Centre de Recerca Matem`atica”, we thought it would be a good opportunity to organize an international conference on Geometric Group Theory, for the second time in Catalonia and in Spain. The organization of this second event shows good international connections of the catalan mathematical community in general and the vitality of Catalonia in this area of mathematics in particular, where the number and quality of Catalan researchers has been signiﬁcantly growing in the last years.

Our main goal in organizing GAGTA has been to offer another opportunity to the international comunity of researchers in Group Theory to meet and discuss recent developments in this interesting and important area of Mathematics. GAGTA is devoted to the study of a variety of topics in Geometric and Combinatorial Group Theory, including asymptotic and probabilistic methods, as well as algorithmic and computational topics related to groups. In addition, highlighting the very recent developments regarding “Group-based Cryptography” and emphasizing this interesting field of applications of Group Theory, one day of the conference will be specifically devoted to this subject: September 4th will be the “Cryptography Day”.

Our ﬁrst words of gratitude are to the University campus holding the conference: the EPSEM, the “Escola Polit`ecnica Superior d’Enginyeria de Manresa”. Both the Direction and the Administration of the EPSEM welcomed this conference with great enthusiasm and helped with a lot of details regarding local arrangements.

Manresa is a middle size city located at the geographical center of Catalonia. It is also located at the very base of Montserrat mountains, the cultural and sentimental heart of Catalonia, too. The proximity to the big city of Barcelona (big in many senses) sometimes eclipses the visibility of the activities promoted and developed in the surrounding smaller cities. Holding GAGTA here on the Manresa University campus is a solid proof of the capability of Manresa and its institutions to hold academic events at the highest international level. In this sense, also non-academic institutions of the city have responded very well to the possibility of running GAGTA here. We 2 warmly thank the “Ajuntament de Manresa” (the City Hall) and the “Caixa Manresa” financial institution for their valuable support as sponsors of the conference, both at the financial and organizational levels. In summary, we are very grateful to the city of Manresa for its warm hospitality towards GAGTA, one of the first international research conferences in any field ever taken place here.

We also wish to acknowledge, with thanks, the support received from other sponsors. Apart from the local ones mentioned in the previous paragraph, we have received a generous support from the following institutions. The “Departament de Matemàtica Aplicada III” (MA3), the host department of the main organizer, and the “Universitat Politècnica de Catalunya” (UPC) as the general holding institution, both collabo- rating at the formal, financial, and practical levels; and the “Oficina de Suport a la Recerca Matemàtica” (OSRM) of the UPC offering valuable technical and human support. Also, our thanks go to the “Societat Catalana de Matemàtiques” (SCM), for its significant help in the framework of giving support to all the events related to Mathematics and held in (or organized from) Catalonia.

Three governments provided generous financial support to the organization of GAGTA through their own promotion programmes for research activities. The “Generalitat de Catalunya”, the Catalan Government, following an established scientific policy even though not always having the appropriate recognition and budget from the Spanish Government; the “Ministerio de Educacióny Ciencia” of the Spanish Government; and the National Science Foundation of the USA, providing a generous support for American participants.

All the sponsors have contributed at diﬀerent levels, and from the organizing committee we want to sincerely thank all of them with the same gratitude. Without any of these contributions the organization would be much more diﬃcult; without all of them, altogether impossible.

In this booklet you will ﬁnd the detailed program of the conference, together with the list of speakers, and the titles and abstracts of their talks. In the program, you will also ﬁnd social activities, including a dinner and a couple of promising cultural visits to Montserrat and Sant Benet, where we hope you can taste and touch a bit of the very rich history, tradition and culture of this country, Catalonia.

We hope you will enjoy your stay in Manresa, wishing it to be very productive from the scientiﬁc point of view, and very pleasant at the same time.

Enric Ventura On behalf of the GAGTA Organizing Committee. Program 4 Program

Thursday, August 31st

(Main Room)

8:30 - 9:30 Registration 9:30 - 10:00 Opening session Josep Burillo 10:00 - 11:00 “Orderings in braided Thompson’s Groups” (p. 14) 11:00 - 11:30 Coﬀee break Martin Lustig 11:30 - 12:30 “R-trees – currents – laminations: a delicate relationship” (p. 19) Gilbert Levitt 12:30 - 13:00 “Growth rates for automorphisms of free groups” (p. 19) Armando Martino 13:00 - 13:30 “Putting a metric on Outer Space” (p. 20)

Sean Cleary 15:30 - 16:00 “Metric properties of braided Thompson’s groups” (p. 15) Sarah Rees 16:00 - 16:30 “Group geodesics: regularity, star-freedom and local testability” (p. 22) 16:30 - 17:00 Coﬀee break Tim Riley 17:00 - 17:30 “Diameter versus ﬁlling length in metric discs and in groups” (p. 24) Kanta Gupta 17:30 - 18:00 “Tree extensions of symmetric groups and automorphisms” (p. 17)

19:00 - 20:00 Reception at Manresa City Hall Program 5

Friday, September 1st

(Main Room)

Mark Sapir 9:00 - 10:00 “TBA” (p. 24) Paul Schupp 10:00 - 11:00 “Mostow-type rigidity as a property of randomness” (p. 25) 11:00 - 11:30 Photo + Coﬀee break Vladimir Shpilrain 11:30 - 12:00 “The word problem: new challenges” (p. 25) Patrick Dehornoy 12:00 - 12:30 “Algorithms for Garside groups” (p. 16) Luis Paris 12:30 - 13:00 “Direct decompositions of Coxeter groups” (p. 22) Anton Klyachko 13:00 - 13:30 “One-relator relative presentations” (p. 19)

Trip to Montserrat 16:00 - 22:30 Concert 6 Program

Saturday, September 2nd

(Main Room)

Oleg Bogopolski 9:00 - 10:00 “On a Magnus property for some one-relator groups” (p. 13) Brian Bowditch 10:00 - 11:00 “The classification of Kleinian groups” (p. 14) 11:00 - 11:30 Coffee break Alain Valette 11:30 - 12:00 “Proper isometric actions on Banach spaces” (p. 26) Gerhard Rosenberger 12:00 - 12:30 “On the surface group conjecture” (p. 24) Luis Ribes 12:30 - 13:00 “On quasifree profinite groups” (p. 23) Denis Osin 13:00 - 13:30 “Large groups and their periodic quotients” (p. 21)

Contributed talks (Main Room, Room S1, Room S2)

MR Fran¸cois Dahmani, “The isomorphism problem for some relatively hyperbolic groups” (p. 31). S1 Elena Bunina, “Elementary equivalence of Chevalley 15:20 - 15:40 groups” (p. 31). S2 Luda Markus-Epstein, “Reading Kurosh decomposition via subgroup graphs” (p. 35). MR Fran¸cois Gautero, “Combination theorem for relatively hyperbolic groups” (p. 33). 15:40 - 16:00 S1 Marius Buliga, “Dilatation structures” (p. 31). S2 Ki Hyoung Ko, “Weighted decompositions of generic braids” (p. 34). Program 7

MR Claas Roever, “TBA” (p. 36). S1 Alexey Muranov, “Finitely generated infinite simple 16:00 - 16:20 groups of infinite commutator width” (p. 35). S2 Daniele Otera, “Some refinements of the (simple) connectivity at infinity” (p. 36). MR Ashot Minasyan, “The SQ-universality and residual properties of relatively hyperbolic groups” (p. 35). S1 Ahmad Erfanian, “Probability of generating simple groups 16:20 - 16:40 PSp(2m,q)” (p. 32). S2 Koji Fujiwara, “Asymptotic dimension of curve graphs” (p. 32). 16:40 - 17:00 Coffee break MR Donghi Lee, “An algorithm that decides translation equivalence” (p. 34). S1 Francesco Russo, “Groups with soluble minimax classes 17:00 - 17:20 of conjugate subgroups” (p. 36). S2 Vladimir Kisil, “Elliptic, parabolic and hyperbolic invariants of cycles” (p. 33). MR Nicholas Touikan, “Two variable equations over free groups” (p. 37). S1 Yuriy Leonov, “Self-similar groups of intermediate growth” 17:20 - 17:40 (p. 38). S2 Uri Weiss, “On biautomaticity of non-homogenous small- cancellation groups” (p. 37). MR Ilya Kazachkov, “Orthogonality and divisibility theories for free partially commutative groups” (p. 33). 17:40 - 18:00 S1 Sergey Zyubin, “Groups acting transitively on a projective line” (p. 38). S2 Ramon J. Flores, “TBA” (p. 32). 18:00 - 18:20 MR Ali Mahdipour Shirayeh, “TBA” (p. 34). 8 Program

Sunday, September 3rd

(Main Room)

Goulnara Arjantseva 9:00 - 10:00 “Uniform properties of finitely generated groups” (p. 13) Alexei Miasnikov 10:00 - 11:00 “Free actions on Lambda-trees” (p. 21) 11:00 - 11:30 Coffee break Mikhail Belolipetsky 11:30 - 12:00 “Counting manifolds and class field towers” (p. 13) Enric Ventura 12:00 - 12:30 “More on the orbit-decidability and the conjugacy problem for free- by-free groups” (p. 26) Edward Turner 12:30 - 13:00 “The density of monomorphisms of free groups” (p. 26) Pascal Weil 13:00 - 13:30 “Algorithmic problems in finitely generated free groups” (p. 27)

Andrew Duncan 15:30 - 16:00 “Partially commutative groups” (p. 16) Ben Fine 16:00 - 16:30 “Discriminating groups” (p. 16) 16:30 - 17:00 Coﬀee break Olga Kharlampovich 17:00 - 17:30 “Equations with parameters in a fully residually free group” (p. 18) Arye Juhasz 17:30 - 18:00 “A Freiheitssatz for whitehead graphs of one-relator groups with small cancellations” (p. 18) Tatiana Smirnova-Nagnibeda 18:00 - 18:30 “TBA” (p. 25)

Visit to “St. Benet” 19:00 - 23:00 Dinner” Program 9

Monday, September 4th

Cryptography Day (Main Room)

Ronald Cramer 9:00 - 9:30 “Algebraic geometric secret sharing schemes and secure multi-party computations over small fields” (p. 41) Consuelo Mart´ınez 9:30 - 10:00 “Non-abelian groups based cryptosystems” (p. 43) Rainer Steinwandt 10:00 - 10:30 “What to expect from a key establishment protocol ?” (p. 44) M. Isabel GonzálezVasco 10:30 - 11:00 “Secure group key establishment: constructions in the standard model” (p. 42) 11:00 - 11:30 Coffee break Boaz Tsaban 11:30 - 12:00 “Generalized length-based attacks on Thompson’s group” (p. 45) Dennis Hofheinz 12:00 - 12:30 “Chosen-ciphertext security: why and how” (p. 43) Jorge Villar 12:30 - 13:00 “Matrix groups over rings and Hensel lifting” (p. 46)

Ben Fine 15:00 - 15:30 “Developing cryptosystems using formal power series rings” (p. 42) Vladimir Shpilrain 15:30 - 16:00 “Public key encryption secure against encryption emulation attack by a computationally unbounded adversary” (p. 44) Alexei Miasnikov 16:00 - 16:30 “Asymptotically dominant properties and subgroup attacks” (p. 44) Alexander Ushakov 16:30 - 17:00 “On length-based attack” (p. 45) 17:00 - 17:30 Coﬀee

Invited Talks

Invited Talks 13

Goulnara Arjantseva (Universitéde Genève) Uniform properties of finitely generated groups

In this talk we discuss various kinds of uniform invariants of a ﬁnitely generated group. Examples are uniform non-amenability, uniform Kazhdan property (T), and uniform embeddability into a Hilbert space.

Mikhail Belolipetsky (Durham University) Counting manifolds and class ﬁeld towers

(Joint work with Alexander Lubotzky). We study the growth rate of lattices of bounded covolume in semi-simple Lie groups. One of the important constructions obtained is based on inﬁnite class ﬁeld towers the existence of which follows from the seminal work of Golod and Shafarevich. This construction provides counterexamples to some previously known conjectures on the growth of lattices and may be also interesting by itself.

Oleg Bogopolski (Dortmund Universit¨at,Novosibirsk State University) On a Magnus property for some one-relator groups

We will say that a group G satisfies the Magnus property if for any two elements u, v ∈ G with the same normal closure, u is conjugate to v or to v−1. In 1930, W. Magnus proved that free groups satisfies this property. We prove that some one-relator groups, including the fundamental groups of closed non-orientable surfaces of genus g = 1, 2 and g > 3 also satisfy Magnus property. The analogous result for orientable surfaces of any finite genus was obtained by the author earlier. Some related problems will be discussed. 14 Invited Talks

Brian Bowditch (University of Southampton) The classiﬁcation of Kleinian groups

We give a brief survey of the main results involved in the recent work of many authors leading to a classification of finitely generated kleinian groups, the resolution of the the tameness and ending lamination conjectures. The latter is closely related to questions regarding the curve complex, and the large scale structure of Teichmüller space. Some of these ideas can be fed back to give new results about the geometry of curve complexes.

Josep Burillo (Universitat Polit`ecnica de Catalunya) Orderings in braided Thompson’s groups

The braided Thompson’s group BV (“braided V ”) has been recently introduced by Brin and Dehornoy, and it is a combination of Thompson’s group V with the braid group B∞ in inﬁnitely many strands. In this talk we will introduce this group and show some of its properties.

An element of Thompson’s group V can be thought of as a triple (T−, π, T+), where T− and T+ are two binary trees (representing subdivisions of the unit interval in successive halves) and π is a permutation on the leaves of the tree, indicating how the subdivisions are mapped into each other. Properly speaking, elements of V are equivalence classes of these triples, where elements are related by subdivision of the leaves in the trees. Then, it is possible to consider the group BV of the triples (T−, b, T+) where b is now a braid in as many strands as the number of leaves in the trees. Multiplication is obtained in a similar way as in V , by ﬁnding representatives where the target tree of the ﬁrst matches the source tree of the second, and then taking the two outside trees and the product of the two braids.

It is known from Brin and Dehornoy that BV is finitely presented, although as it is common in the groups of the Thompson family, they admit infinite presentations which are more regular and are easier to work with. In the talk we will introduce the different presentations of BV .

As it happens in the braid group, BV admits a subgroup of pure braids. The subgroup BF of BV is the subgroup of those triples where the braid is pure. Since the subgroup Invited Talks 15 of V where the permutation is the identity is exactly Thompson’s group F , the subgroup BF of pure braids in BV can be thought of as “braided F ”. We will show (joint work with T. Brady, S. Cleary, and M. Stein), that BF is also ﬁnitely presented.

Ordering properties in braid groups are well-known, and there is an abundance of details in the literature. Namely, it is known that Bn is right-orderable, but not bi- orderable, and its subgroup of pure braids Pn is bi-orderable. We will show (joint work with J. Gonz´alez–Meneses) that the same situation is obtained in the braided Thompson case: BV is right-orderable but not bi-orderable, but its pure subgroup BF is bi-orderable. The key to this result is the study of a “diﬀerent version” of P∞, i.e. a direct limit of pure braid groups where Pn embeds into Pn+1 by splitting a given strand. This group is also bi-orderable and provides the key ingredient of the proof for BF .

Sean Cleary (City College of New York) Metric properties of braided Thompson’s groups

(Joint work with Thomas Brady, Jos´eBurillo and Melanie Stein). Thompson’s groups F , T and V are important examples illustrating a number of pathological group phenomena. There are also braided versions of these groups, for example BV described by Brin and Dehornoy. Fordham developed a method for computing the length of elements of F exactly, which has been used to understand the geometry of the Cayley graph of F . Current understanding of the metric properties of T rely upon estimates of Burillo, Cleary, Stein and Taback and similarly, the metric properties of V rely upon estimates of Birget. Here we describe some of the metric properties of the braided Thompson’s groups. Unlike braid groups, the number of crossings in a standard representative of an element does not give a good estimate for the word length. We give upper and lower bounds for lengths of elements with respect to standard ﬁnite generating sets. 16 Invited Talks

Patrick Dehornoy (Universit´ede Caen) Algorithms for Garside groups

Garside groups generalize braid groups. They share with the latter the existence of eﬃcient computational methods connected with a bi-automatic structure, and, for this reason, they may appear as natural platforms for group-based cryptography. We present solutions to two distinct questions: (i) recognizing whether a given presentation deﬁnes a Garside group, and, if so, (ii) computing in this group by means of the so-called greedy normal form.

Andrew Duncan (Newcastle University) Partially commutative groups

To a graph G is associated a partially commutative group P (alias right-angled Artin group: see I Kazachkov’s abstract) . An orthogonality theory for graphs is developed and used to construct a theory of parabolic and quasiparabolic subgroups of the corresponding groups. This allows a natural description of the centraliser of an arbitrary subset of P and the centraliser lattice of P . As applications we obtain a description of the automorphism group of P .

Benjamin Fine (Fairﬁeld University) Discriminating groups

Discriminating groups were introduced by G.Baumslag, A.Myasnikov and V.Remes- lennikov as an outgrowth of their theory of algebraic geometry over groups. However they have taken on a life of their own and have been an object of a considerable amount of study. In this talk we survey the large array results concerning the class of discriminating groups that have been developed over the past decade. A group G is discriminating if it discriminates any group that it separates. Equvialently G is discriminating if it discriminates its direct square G×G. A group that embeds its direct square is called trivially discriminating abbreviated TD. The theory requires knowing both which groups are discriminating and which are not. Examples of TD groups Invited Talks 17 include many universal type groups such as Higman’s group and Thompson’s group F . A set of examples of non TD groups arose from a class of groups originally studied in a diﬀerent context by B. Neumann. In a negative direction a nilpotent group is proved to be discriminating only if it is abelian. To better capture the axiomatic properties of discriminating groups the class of squarelike groups was introduced by Fine, Gaglione, Myasnikov and Spellman. These are groups which share the same universal theory as their direct squares. The discriminating groups are properly contained in the squarelike groups. Further the squarelike groups have been proved to be axiomatic and in fact are the axiomatic closure of the class of discriminating groups. Finally we consider the relationship of discriminating groups to an older notion of discrimination, which we call varietal discrimination, originally developed by G. Baumslag, B.Neumann,H.Neumann and P. Neumann.

C. Kanta Gupta (University of Manitoba) Tree extensions of symmetric groups and automorphisms

(Joint work with Narain Gupta and Sushchansky). Elements are called primitive in a group G if these elements can be included in a basis of G. Test elements are those elements in a group G ,whenever an endomorphism of G ﬁxes these elements then that endomorphism must be an automorphism.

In my talk I shall give a construction of an automorphism group of a regular rooted tree Tn of valence n which extends on to the tree Tn with the natural action of the symmetric group Sn of degree n with a fixed system of generators consisting of n − 1 transpositions. The group Gn is a subgroup of the automorphism group Aut(Tn). For any n, the group Gn is a finitely automatic permutation group. In the case n = 3, the group G3 contains a subgroup isomorphic to the free product C2 ∗ C2 ∗ C2 of cyclic groups of order 2. R. Grigorchuk posed the question: which groups have representations by automatic permutations ? Brunner and Sidki proved that countable free groups have such representation over a 4 - element alphabet. I shall show that every countable free group is presented by finite automatic permutations over the 3 - element alphabet. Our result improves the statement of Sidki. 18 Invited Talks

Aryeh Juhasz (Technion) A Freiheitssatz for Whitehead graphs of one-relator groups with small cancellations

Let X be a non-empty ﬁnite set and let F be the free group freely generated by X. Let R be a cyclically reduced word and let N be the normal closure of R in F . Denote by supp(R) the the set of all the letters of X which occur in R and R−1.Then a version of the celebrated Freiheitssatz of Magnus states that for every cyclically reduced word U in N, supp(R) is contained in supp(U). Now denote by Supp(R) the set of all the letters of X ∪ X−1 which occur in R. (Thus, if X = {a, b, c}, R = aba−1ba then supp(R) = {a, b} and Supp(R) = {a, a−1, b}. In a recent work we proved that if the symmetric closure of R satisﬁes a small cancellation condition and R is not on a short known list of exceptional words then either Supp(R) is contained in Supp(U), or Supp(R−1) is contained in Supp(U), or both. In terms of Whitehead graphs this means that if we denote by V (R) the set of vertices of the Whitehead graph Wh(R) of R which have valency at least 1 and use the same notation for U, then V (R) is contained in V (U). (Here we consider the “restricted” Whitehead graph in which all edges are simple). In view of the above mentioned version of Magnus’ Freiheitssatz we consider this result as a Freiheitssatz for the vertices of the corresponding Whitehead graphs. In the present talk we prove that under the above assumptions, E(R) is contained in E(U), where E(R) and E(U) are the sets of the edges of Wh(R) and Wh(U), respectively. This is the Freiheitssatz for Whitehead graphs. In particular, it follows from this result that if Wh(R) is 2-connected then Wh(U) also is. We use this result in order to prove a variant of a theorem of V. Shpilrain on the automorphism group of one-relator groups with small cancellations.

Olga Kharlampovich (McGill University) Equations with parameters in a fully residually free group

Solving equations in free groups played one of the key parts in our joint with A. Miasnikov solution of the Tarski problems. It seems, a new structural theory of solutions of equations over groups is taking its shape. I am going to discuss some developments in this area. In particular, I will describe algebraic, transcendental and reducing solutions of equations and show several structural and ﬁniteness results. Invited Talks 19

Anton Klyachko (M.V. Lomonosov Moscow State University) One-relator relative presentations

• Adding one generator and one arbitrary relator to a non-simple group which is either ﬁnite or torsion-free, we always obtain a non-simple group. • Adding two generators and one arbitrary relator to a nontrivial torsion-free group, we always obtain an SQ-universal group.

I shall discuss these and other properties of one-relator relative presentations. Some of these facts can be considered as analogues of well-known theorems on one-relator groups. The proofs are based on geometrical technique such as van Kampen diagrams and the car-crush lemma.

Gilbert Levitt (Universit´ede Caen) Growth rates for automorphisms of free groups

(Joint work with Martin Lustig). Fix an automorphism of a finitely generated free group. Under iteration of the automorphism, any conjugacy class grows as a polynomial times an exponential. Different conjugacy classes may have different growth types. We determine how many different growth types are possible. In particular, we show that there cannot be more than (3n − 2)/4 different exponential growth types, where n is the rank of the free group.

Martin Lustig (Universit´eP. C´ezanne - Aix Marseille III)

R-trees – currents – laminations: a delicate relationship

It is well known that a geodesic lamination L on a closed hyperbolic surface S, provided with a transverse measure µ, determines a small action of π1S on a dual R-tree T (µ), and conversely. The lamination L is non-uniquely ergodic if it admits transverse measures µ0 and µ1 that are (projectively) distinct. In this case, the dual R-trees T (µ1) and T (µ2) are not π1S-equivariently isometric (or homothetic). However, one 20 Invited Talks expects that they are “topologically the same”. A precise answer, to what extend this expectation is justiﬁed, has been given in recent joint work with T. Coulbois and A. Hilion in the more general context of (very) small actions of a free group Fn of ﬁnite rank n on an R-tree T , i.e. of points in the closure CV n of Culler-Vogtmann’s Outer space CVn.

In this more general setting, we also obtain for any such T a “dual” lamination L2(T ), in complete analogy to the geodesic lamination L on the surface S as above, except that L2(T ) is an algebraic lamination, i.e. it is deﬁned entirely in terms of the 2 Gromov boundary ∂FN . For any algebraic lamination L , the object analogous to the transverse measure on L is a current with support L2.

2 Such a current µ with support in L (T ) deﬁnes a dual metric dµ on T , and in the above surface case dµ coincides with the original metric d on T (µ). However, for a 2 large class of very small trees T with uniquely ergodic L (T ) one has d 6= dµ, and indeed, we show that dµ is strongly degenerate for an important class of such trees T . This shows that the question of (non-)unique ergodicity for R-tree actions of Fn is more delicate than expected from the analogy of surface laminations, as it has to be posed independently for both, the metric d and the measure µ.

To clarify (or mistify ?) further the relationship between R-trees and currents, the author has shown in joint recent work with I. Kapovich that there is no map from ∂CVn to the space P Currn of projectivized currents, which is both, contin- uous and equivariant with respect to the natural action of the outer automorphism group Out(Fn). There are many interesting open questions, in particular concerning (i) Kapovich-Nagnibeda’s Out(Fn)-equivariant embedding of CVn into P Currn via Patterson-Sullivan measures, (ii) the natural extension of Thurston’s intersection form for measured laminations to pairs (T, µ) as above, and (iii) the limits set of the Out(Fn)-action on CV n, on P Currn, and on their cartesian product.

Armando Martino (Universitat Polit`ecnica de Catalunya) Putting a metric on Outer Space

(Joint work with Stefano Francaviglila). This talk is about some preliminary results in which we look at a candidate metric for Outer Space, the contractible space on which the Outer automorphism group of a free group acts. Invited Talks 21

Alexei Miasnikov (McGill University) Free actions on Lambda-trees

In this talk I will discuss ﬁnitely generated groups acting freely on Lambda-trees. One of the most intriguing problems in this area (which is due to I.Rips and H.Bass) concerns with the algebraic structure of groups acting freely on arbitrary Lambda- trees. My focus will be on a new approach to this problem via non-Archimedean inﬁnite words and elimination processes.

Denis Osin (City College of New York) Large groups and their periodic quotients

(Joint work with A. Yu. Olshanskii). Using deep results on property (τ) and homology growth, Lackenby proved that if G is a large group, H is a finite index subgroup of G admitting an epimorphism onto a non–cyclic free group, and g is an element of H, then the quotient of G by the normal subgroup generated by gn is large for all but finitely many n. In fact, this result admits a short group theoretic proof based on the Baumslag–Pride theorem about groups with 2 more generators than relations. Moreover, the group theoretic approach allows us to obtain a refinement of the Lackenby Theorem, which can be used to construct new examples of infinite finitely generated periodic groups.

More precisely, we show that for every infinite sequence of primes (p1, p2,...), there exists an infinite finitely generated periodic group Q with descending normal series T Q = Q0 ≥ Q1 ≥ ..., such that i Qi = {1} and Qi−1/Qi is either trivial or finitely generated abelian of exponent pi. For example, using this result we can construct a finitely generated periodic just infinite group Q such that orders of elements of Q are square free and every section of Q is residually finite. In particular, Q is residually finite. Here by a section we mean any quotient group of a subgroup of Q. To the best of our knowledge, no examples of finitely generated infinite periodic groups all of whose sections are residually finite were known until now. 22 Invited Talks

Luis Paris (Universit´ede Bourgogne) Direct decompositions of Coxeter groups

Coxeter groups were introduced by Tits in a 1961 manuscript which has only recently been published, and whose results appeared in the seminal Bourbaki’s book “Groupes et alg`ebres de Lie”, chapters 4, 5 and 6. Coxeter groups have been widely studied, they have many attractive properties, and they form an important source of examples for group theorists. A Coxeter group W is determined by some labeled graph Γ called the Coxeter graph of W (we write W = WΓ). If Γ1,..., Γl are the connected components of Γ, then W is naturally the direct product of the WΓi ’s. The reverse is false: if WΓ is a direct product, then the factors are not necessarily Coxeter groups. However, a recent theorem states that the reverse is actually true if we ignore the ﬁnite factors. Our goal in this talk is to explain this last result.

Sarah Rees (University of Newcastle) Group geodesics: regularity, star-freedom and local testability

The geodesics words in a finitely generated group are known to form a regular set whenever the group is either word hyperbolic or free abelian. For selected generated sets, the same is true for virtually abelian groups, geometrically finite hyperbolic groups, all Coxeter groups, Artin groups of finite type and indeed all Garside groups; this list does not claim to be exhaustive. I report on an investigation to look for connections between algebraic properties of such a group, combinatorial properties of its presentations, the structure of its regular set of geodesics, and the complexity of its word problem. That work is joint work with Gilman, Hermiller and Holt.

Terms such as regularity, star-freedom and local testability will be defined in the talk; each can be shown to have several different disguises (set-theoretic, geometric, or algebraic, in terms of an associated finite semigroup).

We shall see in particular that certain small cancellation conditions on a presentation (which imply word hyperbolicity) force the set of geodesic words to be star-free, that a rather restrictive form of local testability of geodesics implies that the word problem for that group is context-free, and hence characterises virtually free groups, that 1-local testability characterises free abelian groups, and that a group with locally testable geodesics can have only ﬁnitely many conjugacy classes of torsion elements. Invited Talks 23

Luis Ribes (Carleton University) On quasifree proﬁnite groups

(Joint work with Katherine Stevenson and Pavel Zalesskii). A recent characterization of free profinite groups due to Harbater and Stevenson establishes that a profinite group G is free profinite of infinite rank m if and only if

(i) G is projective, and

(ii) whenever one has a diagram

f ? α AB-

where A and B are finite groups, α and f are epimorphisms of profinite groups and α splits, there exist exactly m different epimorphisms λ : G −→ A such that αλ = f.

This builds on other well-known characterizations due to Iwasawa, Mel’nikov and Chatzidakis.

We are interested in profinite groups that satisfy condition (ii) above. For an infinite cardinal m, we define a profinite group G to be m-quasifree if it satisfies condition (ii) above. The following result of Harbater and Stevenson provides naturally arising examples of m-quasifree groups which are not projective, and hence not free profinite.

Theorem. Let k be a field and k ((x, t)) be the fraction field of the power series ring k[[x, t]], where x and t are indeterminates. Let G = Gk((x,t)) be the absolute Galois group of k ((x, t)). Denote by m the cardinality of k ((x, t)). Then G is an m-quasifree profinite group which is not projective.

In our main result we show that open subgroups of m-quasifree groups are m-quasifree. We also provide non-obvious examples of m-quasifree proﬁnite groups. 24 Invited Talks

Tim Riley (Cornell University) Diameter versus ﬁlling length in metric discs and in groups

I will explain recent work with W.P.Thurston concerning duality and the diameters of spanning trees in planar graphs. The concept of the filling length of loops – that is, how long a loop must grow in the course of a null-homotopy – plays a key role and the work has significance for problems of Gromov and Stallings concerning the relationships between filling functions, specifically the Dehn function, the isodiametric function and the filling length function.

Gerhard Rosenberger (Universit¨atDortmund) On the surface group conjecture

Let G = hx1, ..., xn; r = 1i be a non-free one relator group. Assume that each subgroup of ﬁnite index is also a one relator group and, that each subgroup of inﬁnite index is a free group. It is conjectured that G is a surface group. Here we give a partial solution of this conjecture.

Mark Sapir (Vanderbilt University) Cut points in asymptotic cones of groups

(Joint work with C. Drutu, S. Mozes, A. Olshanskii and D. Osin). This is a joint work with C. Drutu, S. Mozes, A. Olshanskii and D. Osin. We characterize groups whose asymptotic cones have cut-points, prove that many non-uniform lattices of semi-simple Lie groups do not have cut-points in any of their asymptotic cones, while many amalgamated products of groups have cut-points in their asymptotic cones. We construct finitely generated groups with all proper subgroups finite (or infinite cyclic), and cut-points in all of their asymtotic cones. We also prove that if a group G has cut- points in its asymptotic cones, then any ”sufficiently large” group cannot have ”too many” pairwise non-conjugate in G homomorphisms into G. This gives a description of non-co-Hopfian relatively hyperbolic groups, relatively hyperbolic groups with infinite Out(G), etc. Invited Talks 25

Paul Schupp (University of Illinois at Urbana-Champaign) Mostow-type rigidity as a property of randomness

(Joint work with Ilya Kapovich and Vladimir Shpilrain). It seems that a very strong Mostow-type rigidity holds for “random groups”. We start with either the free group Fk of rank k ≥ 2 or the modular group M, add a fixed number m of additional defining relators and consider which pairs of such groups are isomorphic as the length n of the defining relators goes to infinity. For quotients G of a fixed group with generating set A the natural associated geometric structure is the Cayley graph Γ(G; A). By “rigidity” in this case we mean that a pair of such groups G1 and G2 should be isomorphic if and only if their associated Cayley graphs Γ(G1; A) and Γ(G2; A) are isometric in the word metric. This property holds strongly generically for one-relator groups and quotients of the modular group M by an arbitrary number m of additional relators. The same should be true for quotients of the free group with any number of defining relators but this is probably very difficult to prove.

Vladimir Shpilrain (City College of New York) The word problem: new challenges

Motivated by recent developments in cryptography, we are going to discuss some previously unattended questions related to the word problem, in particular, how to produce a “random” word equal to 1 in a group given by generators and deﬁning relators.

Tatiana Smirnova-Nagnibeda (University of Geneva)

TBA 26 Invited Talks

Edward Turner (University of Albany) The density of monomorphisms of free groups

(Joint work with Armando Martino and Enric Ventura). We show that for any s and r ≥ 2, the set of monomorphisms from Fs to Fr is dense among the homomorphisms and the set of epimorphisms has density zero. In particular, most homomorphisms from F100 to F2 are (1 − 1) but not onto.

Alain Valette (Universit´ede Neuchˆatel) Proper isometric actions on Banach spaces

A result of Bader-Furman-Gelander-Monod says that every isometric action of a higher rank simple Lie group or lattice on a Lp-space (1 < p < ∞) has a globally ﬁxed point. On the other hand G. Yu proved that any hyperbolic group Γ admits a proper isometric action on `p(Γ × Γ), for p large enough. We will present a result obtained jointly with Y. de Cornulier and R. Tessera: let G be a rank 1 simple Lie group; G admits a proper isometric action on Lp(G) if and only if p is larger than the critical exponent of G.

Enric Ventura (Universitat Polit`ecnica de Catalunya) More on the orbit-decidability and the conjugacy problem for free-by-free groups

(Joint work with O. Bogopolski and A. Martino). Recently, see [1], the conjugacy problem has been solved for free-by-cyclic groups. The proof is very simple and easy, but it uses two deep algorithmic results about free groups, making the actual algorithm provided very complicated (and slow, in the general case). When one turns to consider the conjugacy problem in the family of free-by-free groups, the situation is much more complicated since it is well known the existence of free-by-free groups with unsolvable conjugacy problem (see [3]). However, the arguments used in [1] extend completely to this bigger family, except in only one precise step. This allowed us to ﬁnd and characterize the exact obstruction for a free-by-free group to have solvable conjugacy problem. The main result we shall present is a theorem giving a necessary and suﬃcient condition for the solvability of the conjugacy problem in Invited Talks 27 the family of free-by-free groups; this condition is expressed in terms of what we call “orbit decidability”.

In fact, the result mentioned above is also true in greater generality (see [2]) for a bigger family of group extensions (containing, for example, extensions of free and free abelian groups by torsion-free hyperbolic groups). In the talk, an explicit example of a (free abelian)-by-free group will be exhibit, with unsolvable conjugacy problem; a kind of abelian (and so, much simpler) version of Collins-Miller example.

[1] BMMV Bogopolski, O., Martino, A., Maslakova, O., Ventura, E., Free-by-cyclic groups have solvable conjugacy problem, to appear in Bull. of the London Math. Soc., 8 pages. [2] Bogopolski, O., Martino, A., Ventura, E., The conjugacy problem for some extensions of groups, (work in progress). [3] Collins, D., Miller, C., The conjugacy problem and subgroups of ﬁnite index, Proc. London Math. Soc (3) 34 (1977), 535–556.

Pascal Weil (LaBRI, CNRS and Universit´eBordeaux-1) Algorithmic problems in ﬁnitely generated free groups

We will report recent progress on establishing the algorithmic complexity of two classical problems in ﬁnitely generated free groups:

- deciding whether a subgroup H of a free group F is a free factor of F , - computing a minimum length member of the automorphic orbit of a reduced word w in F (Whitehead minimization problem).

Well-known algorithms show that both problems are decidable, in time polynomial in the length of the input (a set of words generating H, the word w) and exponential in the rank of F . We improve these complexity estimates for both problems.

The results concerning the free factor problem were obtained jointly with Pedro Silva (Porto), and those concerning the Whitehead minimization problem were obtained jointly with Abd´oRoig and Enric Ventura (Barcelona).

Contributed Talks

Contributed Talks 31

Marius Buliga (Institute of Mathematics of the Romanian Academy) Dilatation structures

A dilatation structure is a notion midway between a Lie group and a diﬀerential structure. Many examples of dilatation structures, from metric geometry, iterated functions systems, conical groups, subriemannian geometry, factor spaces of decorated planar forests, will be presented. I shall talk then about Rademacher type theorems for dilatation structures. Depending on available time, I may give arguments supporting the idea that a “fractal group” is a “linear” group associated to a dilatation structure.

Elena Bunina (M.V. Lomonosov Moscow State University) Elementary equivalence of Chevalley groups

Suppose that universal (adjoint) Chevalley groups G(R) and G0(R0) (for example, SLn(R), Sp2n(R), SOn(R), Spinn(R)) are constructed respectively by inﬁnite local rings R and R0 with residue ﬁelds k and k0 of characteristic not equal to 2, and Lie 0 0 0 algebras L and L . Suppose that if L (or L ) has type A1 then k (k ) has more than 3 elements. Then the groups G and G0 are elementarily equivalent if and only if the rings R and R0 are elementarily equivalent and the algebras L and L0 are isomorphic.

This theorem generalizes results of A.I.Maltsev, K.I.Beidar, A.V.Mikhalev.

Fran¸cois Dahmani (Universit´ede Toulouse) The isomorphism problem for some relatively hyperbolic groups

(Joint work with Daniel Groves). The isomorphism problem is the third Dehn’s problem, and ask for an algorithm that decides whether two groups in a given class are isomorphic. In general such a procedure does not exist, and this motivates the search for wide classes in which the problem is decidable. In 1995, Sela published an algorithm for torsion free hyperbolic groups with no cyclic splitting, based on the ﬁniteness of their outer automorphism group. By improving methods on decidability of equations and inequations, we extend his result to the wider class of torsion free 32 Contributed Talks relatively hyperbolic groups with abelian parabolics (without the splitting condition), which also include limit groups, and allows to solve the homeomorphy problem for ﬁnite volume hyperbolic manifolds.

Ahmad Erfanian (Ferdowsi University of Mashhad) Probability of generating simple groups P Sp(2m, q)

Let G be a finite group, and φn(G) be the total number of distinct n-tuples of φn(G) elements of G that generate G. Then, we denote Pn(G) = |G|n as the probability that n randomly chosen elements of G generate G. It is known that when G is a finite non-abelian simple group, then Pn(G) tends to 1 as |G| tends to infinity. In this talk, we will give a lower bound for P2(G), where G is the projective symplectic group P Sp(2m, q). In fact, we prove the following:

88 Theorem. Let G = P Sp(2m, q) and simple. Then P2(G) > 100 for all 2 ≤ m ≤ 5 or m ≥ 10 and q ≥ 2.

Ramon J. Flores (Universidad Carlos III) TBA

Koji Fujiwara (Tohoku University) Asymptotic dimension of curve graphs

Let S be a compact orientable surface of genus g and p punctures. Let C(S) be its curve graph. We show that the asymptotic dimension, asdim, of C(S) is ﬁnite (joint with Bell). We study a geodesic space X with asdim = 1, and show that asdim C(S) > 1 if p = 1 and g > 1 (joint with Whyte). Contributed Talks 33

Fran¸cois Gautero (Universit´eBlaise Pascal) Combination Theorem for relatively hyperbolic groups

We present three distinct results about the combination of relatively hyperbolic groups:

1. An analog, in the setting of relative hyperbolicity, of the Bestvina - Feighn combination theorem for hyperbolic groups. In particular, we treat the case of semi-direct products of relatively hyperbolic groups with free groups.

2. A dynamical characterization of the subgroups to put in the relative part when considering free-by-cyclic groups (this is joint work with M. Lustig).

3. A “converse” to the combination theorem. This relies on a cohomological characterization of strong relative hyperbolicity, following the approach of Gersten in the absolute case (this is joint work with M. Heusener).

Ilya Kazachkov (McGill University) Orthogonality and divisibility theories for free partially commutative groups

To a graph G is associated a partially commutative group P (alias right-angled Artin group). We develop the two tools for the study of free partially commutative groups. It turns out that these give a very powerful language for the theory. In particular, these form the base of the theory of parabolic and quasiparabolic subgroups, centralisers in free partially commutative groups and can be used to obtain a structural description of the automorphism group of P .

Vladimir Kisil (University of Leeds) Elliptic, Parabolic and Hyperbolic Invariants of Cycles

We present foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of geometry and analysis based on the representation theory of the SL group. The talk is concentrated around the description of appropriate geometric 34 Contributed Talks invariants. The principal role is played by Cliﬀord algebras of matching types and a generalisation of the Fillmore–Springer–Cnops construction which describe cycles as points in the extended space. Interesting non-local and non-symmetrical behaviour closed to geometry of non-commutative space is observed. Full text is at: http://arxiv.org/abs/math.CV/0512416.

Ki Hyoung Ko (Korea Advanced Institute of Science and Technology) Weighted decompostions of generic braids

(Joint work with Jang Won Lee). We discuss the usefulness of weighted decompos- tion invented by Garside and Thurston to analyze various behaviour of generic braids that are either products of randomly chosen permutation braids or pseudo-Anosov braids.

Donghi Lee (Pusan National University) An algorithm that decides translation equivalence

Let Fn be a free group of rank n ≥ 2. Two elements u, v in Fn are said to be translation equivalent in Fn if the cyclic length of φ(u) equals the cyclic length of φ(v) for every automorphism φ of Fn. We will prove that translation equivalence is algorithmically decidable in F2. We will also discuss other interesting problems that can be solved with similar techniques.

Ali Mahdipour Shirayeh (Iran University of Science and Technology)

TBA Contributed Talks 35

Luda Markus-Epstein (Technion) Reading Kurosh Decomposition via Subgroup Graphs

A well known result of Stallings says that a finitely generated subgroup of a free group can be canonically represented by a finite labeled graph. It turns out that the same happens for finitely generated subgroups of amalgams of finite groups. Namely, they can be effectively represented by finite canonical graphs. These graphs contain all the essential information about the subgroups, which enables one to use them in order to solve various algorithmic problems. An example of such applications in the case of finitely generated subgroups of free products of finite groups is an easy reading procedure of Kurosh Decomposition via subgroup graphs.

Ashot Minasyan (University of Geneva) The SQ-universality and residual properties of relatively hyperbolic groups

(Joint work with Goulnara Arjantseva and Denis Osin). We apply methods of generalized small cancellation theory to study the residual properties of relatively hyperbolic groups.

Let G be a non-elementary group hyperbolic relatively to a collection of proper subgroups. In this case we show that any countable group is embeddable into some quotient of G, i.e., G is SQ-universal. We also prove that if G is ﬁnitely presented, then it possesses a lot of ﬁnitely presented quotients having “bad” group-theoretic properties.

Alexey Muranov (Vanderbilt University) Finitely generated inﬁnite simple groups of inﬁnite commutator width

In 1951, Oystein Ore conjectured that all elements in every non-abelian ﬁnite simple group are commutators. In terms of commutator width, the conjecture is that the commutator width of every non-abelian ﬁnite simple group is 1. This question still remains open. 36 Contributed Talks

In 1999, Valerij Bardakov posed the following question in the Kourovka Notebook: Does there exist a (ﬁnitely presented) simple group of inﬁnite commutator width?

It was shown by Jean-Marc Gambaudo and Etienne´ Ghys in 2004 that there are infinite simple groups of surface diffeomorphisms for which the commutator width is infinite.

I show that finitely generated infinite simple groups of infinite commutator width, as well as of large finite commutator width, can be constructed using the small- cancellation theory.

Daniele Otera (Universitàdi Palermo) Some refinements of the (simple) connectivity at infinity

(Joint work with L. Funar). In the talk we shall be interested in some topological notions of groups, in particular: the simple connectivity at inﬁnity, its growth and the end-depth. A group is simply connected at inﬁnity (sci) if its Cayley complex is sci. The sci growth, V1, is a function measuring the rate of vanishing of the sci. The end-depth is the 0-dimensional analogous of the sci growth. We shall show that these are quasi-isometry invariants of groups and that for many groups these functions are linear, by giving also some relationships for amalgamated free products.

Claas Roever (NUI Lalway)

TBA

Francesco Russo (University of Wuerzburg) Groups with soluble minimax classes of conjugate subgroups

A famous Neumann’s Theorem characterizes the groups in which each subgroup has finitely many conjugates only as central-by-finite groups (see [3]). Given a class of groups X, a group G is said to have X classes of conjugate subgroups if G/coreG(NG(H)) ∈ X for each subgroup H of G. If X is the class of all finite groups, each group with finite classes of conjugate subgroups is central-by-finite. Contributed Talks 37

If X is a class of soluble groups Which satisfy the maximal or the minimal condition on subgroups, it is possible to improve Neumann’s Theorem. A description of groups with polycyclic-by-ﬁnite classes of conjugate subgroups is given in [1] and a description of groups with Chernikov classes of conjugate subgroups is given in [2]. The next natural step is the study of the groups with soluble minimax classes of conjugate subgroups.

We ﬁnd that a group G, which has soluble minimax classes of conjugate subgroups, can be characterized in terms of G/Z(G).

[1] Kurdachenko, L., Otal, J., Soules, P., Polycyclic-by-ﬁnite conjugate classes of subgroups, Comm. Algebra 32 (2004), 4769–4784. [2] Kurdachenko, L., Otal, J., Groups with Chernikov classes of conjugate subgroups, J. Group Theory 8 (2005), 93–108. [3] Neumann, B.H., Groups with ﬁnite classes of conjugate subgroups, Math. Z. 63 (1955), 76–96.

Nicholas Touikan (McGill University) Two variable equations over free groups

(Joint work with Derrick Chung). A description of the solution sets of two variable equations over a free group was given more than two decades ago by Ozhigov and Hmelevskii; unfortunately the proofs involved as well as the descriptions of these solution sets were excessively complicated. Using results from more recent works of Kharlampovich and Miasnikov, we recover this description of the solution sets of two variable equations. The main advantage of our new approach is that it is much simpler and provides an illuminating illustration of what is going on in modern theory of equations over free groups.

Uri Weiss (Technion) On biautomaticity of non-homogenous small-cancellation groups

We deﬁne the notion of V (6) presentations that naturally generalize C(4)&T (4) and C(6) small-cancellation presentations. V (6) groups have van Kampen diagrams that 38 Contributed Talks locally look like the van Kampen diagrams of C(4)&T (4) and C(6) groups. We show that for presentations with suitable order on words, a biautomatic structure can be constructed through minimal elements of the order. Then, such order is explicitly given for V (6) presentations where every piece is of length one.

Yuriy Leonov (Odessa Academy of Telecommunications) Self-similar groups of intermediate growth

We study the growth of self-similar groups. A new conditions of subexponential- ity was obtained. It allows to prove that some well-known self-similar groups have intermediate growth.

Sergey Zyubin (Tomsk Polytechnic University) Groups acting transitively on a projective line

Let K be a (commutative or skew) field. In the first part of problem 11.70 in the Kourovka Notebook P. Neumann and C. Praeger ask to find all transitive subgroups 1 of the group P GL2(K) acting on the projective line P (K).

The following result gives a partial answer to this problem.

Theorem. Let K be a locally ﬁnite ﬁeld and G be a subgroup of P GL2(K). If G acts transitively on the projective line P 1(K) then the only following cases are possible:

1. G = P GL2(K);

2. G = PSL2(K);

3. G is diagonalizable over a quadratic extension of the ﬁeld K;

4. G has a subgroup of index 2 that is diagonalizable over a quadratic extension of the ﬁeld K.

We also give a clear description of the subgroup G in the third and the forth cases. Cryptography Day Talks

Cryptography Day Talks 41

Ronald Cramer (CWI and Leiden University) Algebraic geometric secret sharing schemes and secure multi-party Computations over Small Fields

(Joint work with Hao Chen). We introduce algebraic geometric techniques in secret sharing and in secure multi-party computation (MPC) in particular. The main result is a linear secret sharing scheme (LSSS) defined over a finite field Fq, with the following properties.

1. It is ideal. The number of players n can be as large as #C(Fq), where C is an algebraic curve C of genus g deﬁned over Fq. 2. It is quasi-threshold: it is t-rejecting and t+1+2g-accepting, but not necessarily t + 1-accepting. It is thus in particular a ramp scheme. High information rate can be achieved.

3. It has strong multiplication with respect to the t-threshold adversary structure, 1 4 if t < 3 n − 3 g. This is a multi-linear algebraic property on an LSSS facilitating zero-error multi-party multiplication, unconditionally secure against corruption by an active t-adversary.

4. The finite field Fq can be dramatically smaller than n. This is by using algebraic curves with many Fq-rational points. For example, for each small enough , there is a finite field Fq such that for infinitely many n there is an LSSS over 1 1 Fq with strong multiplication satisfying ( 3 − )n ≤ t < 3 n. 5. Shamir’s scheme, which requires n > q and which has strong multiplication for 1 t < 3 n, is a special case by taking g = 0.

Now consider the classical (“BGW”) scenario of MPC unconditionally secure (with 1 zero error probability) against an active t-adversary with t < 3 n, in a synchronous n-player network with secure channels. By known results it now follows that there exist MPC protocols in this scenario, achieving the same communication complexities in terms of the number of ﬁeld elements exchanged in the network compared with known Shamir-based solutions. However, in return for decreasing corruption tolerance by a small -fraction, q may be dramatically smaller than n. This tolerance decrease is unavoidable due to properties of MDS codes. The techniques extend to other models of MPC. Results on less specialized LSSS can be obtained from more general coding theory arguments. 42 Cryptography Day Talks

Published in Proceedings of 26th Annual IACR CRYPTO, Santa Barbara, Ca., USA, Springer LNCS.

Benjamin Fine (Fairﬁeld University) Developing cryptosystems using formal power series rings

(Joint work with Gilbert Baumslag and Gerhard Rosenberger). Recently there has been a great deal of work on noncommutative algebraic cryptography. This involves the use of noncommutative algebraic objects as the platforms for encryption systems. Most of this work, such as the Anshel-Anshel-Goldfeld scheme, the Ko-Lee scheme and the Baumslag-Fine-Xu modular group scheme use nonabelian groups as the basic algebraic objects. Some of these encryption methods have been successful and some have been broken. It has been suggested that at this point further pure group theoretic research, with an eye towards cryptographic applications, is necessary. In the present study we attempt to extend the class of noncommutative algebraic objects to be used in cryptography. In particular we explore several diﬀerent methods to use a formal power series ring Rhhx1, ..., xnii in noncommuting variables x1, ..., xn as a base to develop cryptosystems. Although R can be any ring, we have in mind formal power series rings over the rationals. We use in particular a result of Magnus that a ﬁnitely generated free group has a faithful representation in a quotient of the formal power series ring in noncommuting variables.

Mar´ıa Isabel Gonz´alez Vasco (Universidad Rey Juan Carlos) Secure group key establishment: constructions in the standard model

(Joint work with Jens-Matthias Bohli and Rainer Steinwandt). Several group key establishment protocols have been proposed that are both eﬃcient and probably secure. However, most of these constructions rely on the existence of idealized hash functions, i.e., are presented in the so-called random oracle model. Moreover, many constructions for authenticated group key establishment rely on rather strong assumptions, like the availability of shared high-entropy secrets or of a PKI for signing messages.

We prove that secure authenticated group key establishment can be achieved with rather minimal assumptions, namely password-based authentication and the common Cryptography Day Talks 43 reference string model. In addition, we discuss the possibility of designing group key establishment protocols in the standard model using (non-abelian) group theoretical problems as a base.

Dennis Hofheinz (Centrum voor Wiskunde en Informatica, Amsterdam) Chosen-ciphertext security: why and how

This talk reﬂects security notions for encryption schemes, focusing on the property of chosen-ciphertext security. This is a notion of security in which an adversary is allowed to request decryptions of arbitrary ciphertexts of his choice. We motivate this notion, give examples of secure and insecure schemes, and show general design strategies to achieve this strong notion of security.

Consuelo Mart´ınez (Universidad de Oviedo) Non-abelian groups based cryptosystems

One of the first proposals of a IND-CCA cryptographic scheme is due to Cramer and Shoup. They gave also a general construction that allows the design of group theoretic schemes that can be proved to be IND-CCA secure. The main ingredients in their construction are: projective hash families, subset membership problems and universal hash proof systems (as a bridge between the previous two ingredients). A similar proposal in a non-abelian context was given by González-Vasco, Mart´ınez, Steinwandt and Villar. One of main differences beween both constructions is that, instead of using group systems for deriving projective hash families (as Cramer and Shoup did in the abelian case), in the non-abelian case automorphism group systems are used. However no new different subset membership problems were identified in the non-abelian proposal. Here we will approach to nonabelian subset membership problems, opening new possibilities to the above mentioned framework designed in a context of non-abelian groups. 44 Cryptography Day Talks

Alexei Miasnikov (McGill University) Asymptotically dominant properties and subgroup attacks

In this talk I will focus on asymptotic properties of subgroups of inﬁnite groups. It turns out that “generic subgroups” quite often have very speciﬁc properties. Thus, a randomly chosen subgroup is subject to various subgroup attacks based on these properties. I will discuss how one could avoid such attacks by choosing the subgroups carefully.

Vladimir Shpilrain (City College of New York) Public key encryption secure against encryption emulation attack by a computationally unbounded adversary

(Joint work with Denis Osin). The purpose of this talk is to explain why, contrary to a prevalent opinion, public key encryption can be secure against “encryption emulation” attacks by a computationally unbounded adversary, with one reservation: a legitimate party decrypts correctly with probability that can be made arbitrarily close to 1, but not equal to 1.

Rainer Steinwandt (Florida Atlantic University) What to expect from a key establishment protocol ?

(Joint work with Jens-Matthias Bohli, Benjamin Glas and Mar´ıa Isabel Gonz´alez Vasco). The talk addresses the issue of formalizing the security requirements for (group) key establishment. At this, also the problem of malicious insiders is addressed, which in particular for key establishments with n 2 participants is of interest.

As an example, a 2-round group key establishment is discussed, whose security proof builds on the random oracle model and assumes the availability of globally veriﬁable signatures. The mathematical assumption underlying the security proof is geared towards non-abelian groups, but can be instantiated with the Computational Diﬃe- Hellman assumption. Cryptography Day Talks 45

Boaz Tsaban (Weizmann Institute of Science) Generalized length-based attacks on Thompson’s group

(Joint work with Dima Ruinskiy and Adi Shamir). The idea of using length functions for solving “random” equations in noncommutative groups is not new. Hughes and Tannenbaum have indicated its potential applicability for attacking cryptosystems that use random subgroups of the braid group as their platform. Garber, Kaplan, Teicher, Tsaban, and Vishne, showed that a memory-based extension of this approach is rather successful for the mentioned groups, see: http://dx.doi.org/doi:10.1016/j.aam.2005.03.002

However, this approach relies heavily on the fact that the underlying group is close to being free. In 2004, Shpilrain and Ushakov suggested using Thompson’s group as a platform for a cryptosystem. This group is far from being free, and therefore poses a real challenge for the length-based approach. Indeed, an initial estimation of a length-based attack is reported, for which the success rate was zero.

We introduce various improvements on the length-based approach which make it more suitable for “far from free” groups, and show that these allow a cryptoanalysis of the Shpilrain-Ushakov Thompson’s group cryptosystem with signiﬁcant success rates. (This applies to the system and the parameters as suggested in the paper. Iterations may foil the present attacks.)

These results suggest that length-based cryptoanalysis is to cryptosystems based on (noncommutative) combinatorial group theory like diﬀerential and linear cryptoanalysis are to classical symmetric ciphers. No cryptosystem based on noncommutative combinatorial group theory can be considered secure before tested against our attacks.

Alexander Ushakov (Stevens Institute of Technology) On length-based attack

1. We show (experimentally) that accurately designed length-based attack can crack a random instance of the multiple conjugacy search problem in Braid Groups (for certain parameter values), argue that completely random choice of public/private information in AAG key exchange protocol leads to weak keys. 46 Cryptography Day Talks

2. We propose a method for key generation resistant against this and several other known attacks.

Jorge Villar (Universitat Polit`ecnica de Catalunya) Matrix groups over rings and Hensel lifting

In this talk, we analyze the hardness of the discrete logarithm problem on matrix groups over finite commutative rings and its relation to the traditional discrete logarithm problem, over a finite field. Some considerations about the conjugacy problem in such groups are also done. List of Participants

List of Participants 49

1. Maxwell Agyei (University College of Mansa) [email protected], 2. Taofeek Ajibola Alabi (University of Ilorin, Kwara) [email protected], 3. Montserrat Alsina (Universitat Politècnica de Catalunya) [email protected], 4. Alili Amar alili [email protected], 5. Yago Antol´ınPichel (Universitat Autònomade Barcelona) [email protected], 6. Javier Aramayona (University of Warwick) [email protected], 7. Goulnara Arjantseva (Universitéde Genève) [email protected], 8. Nguyen Aude (UniversitéLibre de Bruxelles) [email protected], 9. Llu´ısBacardit i Carrasco (Universitat Autònoma de Barcelona) [email protected], 10. Borja Balle (Universitat Politècnica de Catalunya) [email protected], 11. Kazeem Balogun (Federal University of Technology Akure) [email protected], 12. Udo Baumgartner (University of Newcastle) [email protected], 13. Mikhail Belolipetsky (Durham University) [email protected], 14. Oleg Bogopolski (Novosibirsk University) [email protected], 15. Brian Bowditch (University of Southampton) [email protected], 16. Marius Buliga (Institute of Mathematics of the Romanian Academy) [email protected], 17. Elena Bunina (M.V. Lomonosov Moscow State University) [email protected], 18. Josep Burillo (Universitat Politècnica de Catalunya) [email protected], 19. Colin Campbell (University of St. Andrews) [email protected], 20. Montserrat Casals Ruiz (McGill University) 50 List of Participants

[email protected], 21. Laura Ciobanu (University of Auckland) [email protected], 22. Sean Cleary (City College of New York) [email protected], 23. Thierry Coulbois (UniversitéAix-Marseille III) [email protected], 24. Ronald Cramer (CWI and Leiden University) [email protected], 25. Fran¸cois Dahmani (Universitéde Toulouse) [email protected], 26. Yves de Cornulier (EPF Lausanne) [email protected], 27. Patrick Dehornoy (Universitéde Caen) [email protected], 28. Warren Dicks (Universitat Autònomade Barcelona) [email protected], 29. Andrew Duncan (Newcastle University) [email protected], 30. Ahmad Erfanian (Ferdowsi University of Mashhad) [email protected], 31. Benjamin Fine (Fairfield University) [email protected], 32. Francesc Fité(Universitat Politècnica de Catalunya) [email protected], 33. Ramon J. Flores (Universidad Carlos III) [email protected], 34. Koji Fujiwara (Tohoku University) [email protected], 35. Fran¸cois Gautero (UniversitéBlaise Pascal) [email protected], 36. Mar´ıaIsabel González Vasco (Universidad Rey Juan Carlos) [email protected], 37. Juan González-Meneses (Universidad de Sevilla) [email protected], 38. C. Kanta Gupta (University of Manitoba) [email protected], 39. Luc Guyot (Universitéde Génève) [email protected], List of Participants 51

40. Arnaud Hilion (UniversitéAix-Marseille 3) [email protected], 41. Dennis Hofheinz (Centrum voor Wiskunde en Informatica) [email protected], 42. Momodou Jaiteh (Muslim Hands Bus. Computer Tr. Institute) [email protected], 43. Eric Jaligot (CNRS - UniversitéLyon 1) [email protected], 44. Aryeh Juhasz (Technion) [email protected], 45. Ilya Kapovich (University of Illinois at Urbana-Champaign) [email protected], 46. Aditi Kar (University of Nebraska at Lincoln) [email protected], 47. Ilya Kazachkov (McGill University) [email protected], 48. Olga Kharlampovich (McGill University) [email protected], 49. Vladimir Kisil (University of Leeds) [email protected], 50. Anton Klyachko (M.V. Lomonosov Moscow State University) [email protected], 51. Ki Hyoung Ko (Korea Advanced Institute of Science and Technology) [email protected], 52. Natalia Kopteva (Universitéde Provence) [email protected], 53. Donghi Lee (Pusan National University) [email protected], 54. JörgLehnert (University of Frankfurt) [email protected], 55. Yuriy Leonov (Odessa Academy of Telecommunications) leonov [email protected], 56. Gilbert Levitt (Universitéde Caen) [email protected], 57. Martin Lustig (UniversitéP. Cézanne - Aix Marseille III) [email protected], 58. Ali Mahdipour Shirayeh (Iran University of Science and Technology) mahdi [email protected], 59. Luda Markus-Epstein (Technion) 52 List of Participants

[email protected], 60. Consuelo Mart´ınez (Universidad de Oviedo) [email protected], 61. Armando Martino (Universitat Politècnica Catalunya) [email protected], 62. Alexei Miasnikov (McGill University) [email protected], 63. Ashot Minasyan (University of Geneva) [email protected], 64. Alexey Muranov (Vanderbilt University) [email protected], 65. Kei Nakamura (University of California, Davis) [email protected], 66. Denis Osin (City College of New York) [email protected], 67. Daniele Otera (Universitàdi Palermo) [email protected], 68. Luis Paris (Universitéde Bourgogne) [email protected], 69. Emmanuel Phebirih (Technical Institute of Art) [email protected], 70. Matthieu Picantin (LIAFA UniversitéParis 7) [email protected], 71. Jacqui Ramagge (University of Newcastle) [email protected], 72. Sarah Rees (University of Newcastle) [email protected], 73. Luis Ribes (Carleton University) [email protected], 74. Tim Riley (Cornell University) [email protected], 75. Claas Roever (NUI Galway) [email protected], 76. AbdóRoig (Universitat Politècnicade Catalunya) [email protected], 77. Gerhard Rosenberger (Universität Dortmund) [email protected], 78. Alina Rull (University of Zurich) [email protected], List of Participants 53

79. Francesco Russo (University of Wuerzburg) [email protected], 80. Muhammad Sarwar Saeed (Heriot-Watt University) [email protected], 81. Mark Sapir (Vanderbilt University) [email protected], 82. Ritumoni Sarma (Harish-Chandra Research Institute) [email protected], 83. Marcin Sawicki (Warsaw University) [email protected], 84. Paul Schupp (University of Illinois at Urbana-Champaign) [email protected], 85. Pascal Schweitzer (Max Plank Institut for Computer Science Saarbrücken) [email protected], 86. Kenneth Shackleton (Tokyo Institute of Technology) [email protected], 87. Vladimir Shpilrain (City College of New York) [email protected], 88. Tatiana Smirnova-Nagnibeda (University of Geneva) [email protected], 89. Rainer Steinwandt (Florida Atlantic University) [email protected], 90. Simon Thomas (Rutgers University) [email protected], 91. Nicholas Touikan (McGill University) [email protected], 92. Boaz Tsaban (The Weizmann Institute of Science) [email protected], 93. Edward Turner (University of Albany) [email protected], 94. Alexander Ushakov (Stevens Institute of Technology) [email protected], 95. Alain Valette (Universitéde Neuchâtel) [email protected], 96. Enric Ventura (Universitat Politècnica Catalunya) [email protected], 97. Jorge Villar (Universitat Politècnica de Catalunya) [email protected], 98. Pascal Weil (LaBRI, CNRS) 54 List of Participants

[email protected], 99. Uri Weiss (Technion) [email protected], 100. Gerald Williams (University of Kent) [email protected], 101. Asli Yaman (CRM) [email protected], 102. El From Youssef (Universit´eCadi Ayyad, FSSM) [email protected], 103. Roland Zarzycki (University of Wroclaw) [email protected], 104. Fabio Zuddas (Universit`adi Cagliari) [email protected], 105. Eric Zupunski (University of Michigan) [email protected], 106. Sergey Zyubin (Tomsk Polytechnic University) [email protected], Contents

Program 3

Thursday, August 31st ...... 4

Friday, September 1st ...... 5

Saturday, September 2nd ...... 6

Sunday, September 3rd ...... 8

Monday, September 4th ...... 9

Invited Talks 11

Goulnara Arjantseva ...... 13

Mikhail Belolipetsky ...... 13

Oleg Bogopolski ...... 13

Brian Bowditch ...... 14

Josep Burillo ...... 14

Sean Cleary ...... 15

Patrick Dehornoy ...... 16

Andrew Duncan ...... 16

Benjamin Fine ...... 16

55 56 Contents

C. Kanta Gupta ...... 17

Aryeh Juhasz ...... 18

Olga Kharlampovich ...... 18

Anton Klyachko ...... 19

Gilbert Levitt ...... 19

Martin Lustig ...... 19

Armando Martino ...... 20

Alexei Miasnikov ...... 21

Denis Osin ...... 21

Luis Paris ...... 22

Sarah Rees ...... 22

Luis Ribes ...... 23

Tim Riley ...... 24

Gerhard Rosenberger ...... 24

Mark Sapir ...... 24

Paul Schupp ...... 25

Vladimir Shpilrain ...... 25

Tatiana Smirnova-Nagnibeda ...... 25

Edward Turner ...... 26

Alain Valette ...... 26

Enric Ventura ...... 26

Pascal Weil ...... 27 Contents 57

Contributed Talks 29

Marius Buliga ...... 31

Elena Bunina ...... 31

Fran¸coisDahmani ...... 31

Ahmad Erfanian ...... 32

Ramon J. Flores ...... 32

Koji Fujiwara ...... 32

Fran¸coisGautero ...... 33

Ilya Kazachkov ...... 33

Vladimir Kisil ...... 33

Ki Hyoung Ko ...... 34

Donghi Lee ...... 34

Ali Mahdipour Shirayeh ...... 34

Luda Markus-Epstein ...... 35

Ashot Minasyan ...... 35

Alexey Muranov ...... 35

Daniele Otera ...... 36

Claas Roever ...... 36

Francesco Russo ...... 36

Nicholas Touikan ...... 37

UriWeiss ...... 37

Yuriy Leonov ...... 38

Sergey Zyubin ...... 38 58 Contents

Cryptography Day Talks 39

Ronald Cramer ...... 41

Benjamin Fine ...... 42

Mar´ıaIsabel Gonz´alezVasco ...... 42

Dennis Hofheinz ...... 43

Consuelo Mart´ınez ...... 43

Alexei Miasnikov ...... 44

Vladimir Shpilrain ...... 44

Rainer Steinwandt ...... 44

Boaz Tsaban ...... 45

Alexander Ushakov ...... 45

Jorge Villar ...... 46

List of Participants 47

Contents 55