# Program

## Abstracts

By order of appearance

Tatiana Smirnova Nagnibeda (U Geneva, Switzerland)
Spectral theory of groups and group actions
Abstract: We shall discuss spectral properties of Laplacians (or Markov operators of the random walks) on Cayley graphs of finitely generated groups and on a more general class of graphs called Schreier graphs, corresponding to non-free group actions. Our main class of examples are self-similar groups and self-similar actions. In some cases interesting parallels appear with the spectral theory of Schroedinger operators on quasi-crystals.

Nir Avni (Northwestern)
First order rigidity, biinterpretation, and higher rank lattices.
Abstract: In many contexts, there is a dichotomy between lattices in Lie groups of rank one and lattices in Lie groups of higher rank, where the two classes behave in qualitatively different ways. I will talk about a manifestation of this dichotomy in Model Theory. Based on joint works with Alex Lubotzky and Chen Meiri.

Zlil Sela (Hebrew U)
Basic conjectures and preliminary results in non-commutative algebraic geometry
Abstract: Algebraic geometry studies the structure of sets of solutions (varieties) over fields and commutative rings. Starting in the early 1960's ring theorists (Cohn, Bergman and others) have tried to study the structure of varieties over non-commutative rings (notably free associative algebras). The lack of unique factorization that they tackled and studied in detail, and the pathologies that they were aware of, prevented any attempt to prove or even speculate what can be the properties of such varieties.
Using techniques and concepts from geometric group theory and from low dimensional topology, we formulate concrete conjectures on the structure of these varieties, and prove preliminary results in the direction of these conjectures.

Boaz Tsaban (Bar Ilan)
Nonabelian Cryptology and Algebraic Span Cryptanalysis
Abstract: I will provide an introduction to public key cryptography based on nonabelian groups, and to a powerful cryptanalytic method devised recently. I will conclude with a brief discussion of the future of nonabelian cryptology, in light of this method.
This talk is partially based on a joint work with Adi Ben-Zvi and Arkadius Kalka

Oleg Bogopolski (Heinrich Heine U, Dusseldorf)
Algebraically and verbally closed subgroups of groups: Equations in acylindrically hyperbolic groups
Abstract: We describe solutions of certain equations in acylindrically hyperbolic groups (AH-groups). Using this description and certain test words in AH-groups, we study the verbal closedness of AH-subgroups in groups, see [1] below.
As a corollary, we solve a problem of Myasnikov and Roman'kov from [2]. Note that verbal, algebraic, existential, and elementary types of closedness of subgroups in groups are being intensively studied in GGT and model theory. Let F(X) be the free group with infinite countable basis x1,x2,... Recall: A subgroup H of a group G is called verbally closed in G if for any word w(x1,...,xn) in F(X) and any element h in H the equation w(x1,...,xn)=h has a solution in G if and only if it has a solution in H.
MAIN THEOREM. Let G be a finitely presented group and H a finitely generated subgroup of G. Suppose that H is acylindrically hyperbolic and does not have nontrivial finite normal subgroups. Then H is verbally closed in G if and only if H is a retract of G.
The condition that G is finitely presented and H is finitely generated can be replaced by the condition that G is finitely generated over H and H is equationally noetherian.
COROLLARY 1. (Solution of Problem 5.2 from [2]). Verbally closed subgroups of torsion-free hyperbolic groups are retracts.
Recall: A subgroup H of a group G is called algebraically closed in G if for any finite system of equations with coefficients in H, this system has a solution in G if and only if it has a solution in H.
Surprisingly, under some mild assumption, the notions of verbal closedness, algebraic closedness, and retractness become the same after taking free products. More precisely:
COROLLARY 2. Suppose that H is a finitely generated subgroup of a finitely presented group G. Then for any nontrivial finitely presented group A with |A|>2 the following conditions are equivalent.
(1) H*A is algebraically closed in G*A.
(2) H*A is verbally closed in G*A.
(3) H*A is a retract of G*A.
Note that for A=1, these conditions are not equivalent in general.
References: [1] O. Bogopolski, Equations in acylindrically hyperbolic groups and verbal closedness, 2018. [2]
A.G. Myasnikov and V. Roman'kov (Verbally closed subgroups of free groups, J. of Group Theory, 17, no. 1 (2014), 29-40).

Cannon-Thurston maps for CAT(0) groups with isolated flats
Abstract: Given a hyperbolic 3-manifold that fibers over a circle with fiber a surface, Cannon and Thurston related the boundary of the surface group with the boundary of the 3-manifold group. Motivated to find similar phenomenon for CAT(0) groups, we show that Cannon-Thurston maps do not exist for certain infinite index, normal subgroups of a CAT(0) group with isolated flats. This is joint work with Benjamin Beeker, Mathew Cordes, Giles Gardam and Emily Stark.

Word problem solutions by finite state automata
Abstract: In this talk I will discuss several ways to solve the word problem for groups by finite automata, including automatic and autostackable structures, along with geometric and topological views of these properties. Applications will be given to a variety of families of groups, including Thompson's group F, 3-manifold groups, graphs of groups, and Artin groups.
Based on joint projects with N. Corwin, G. Golan, A. Johnson, and Z. Sunic, with M. Brittenham and T. Susse, and with D. Holt, S. Rees, and T. Susse.

Alex Lubotzky (Hebrew U)
Groups approximation, stability and high dimensional expanders
Abstract: Several well-known open questions, such as: "are all groups sofic or hyperlinear?", have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms.
We answer some of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L2) norm or by the Schachten p-norms.
The strategy is via the notion of "stability" : some 2nd cohomology vanishing is proven to imply stability and some high dimensional expanders are used to give existance of non residually finite groups ( central extensions of some lattices in p-adic Lie groups) which are stable and hence cannot be approximated. All notions will be explained.
Based on joint works with M. De Chiffre, L. Glebsky and A. Thom and with I. Oppenheim.

Arie Levit (Yale)
Surface groups are flexibly stable
Abstract: We show that surface groups are flexibly stable in permutations. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.
This is a joint work with Nir Lazarovich and Yair Minsky.

William Menasco (Buffalo)
Distance and intersection number in the curve complex
Abstract: Let Sg be a closed oriented surface of genus g≥ 2 and 𝒞1(Sg) be its complex curvevertices are homotopy classes of essential simple closed curves with two vertices sharing an edge if they have disjoint representatives. It is known that 𝒞1(Sg) is path connected, and the distance, d(α, β), between two vertices α, β ∈ 𝒞1(S) is just the minimal count of the number of edges in an edge-path between α and β. One can also consider, i(α,β), the minimal intersection between curve representatives of α and β. This talk will discuss how i(α,β) grows as d(,) grows.
This is joint work with Dan Margalit and additionally features joint work with Joan Birman and Dan Margalit.

Elizabeth Field (Illinois)
Trees, dendrites, and the Cannon-Thurston map
Abstract: When 1 -> H -> G -> Q -> 1 is a short exact sequence of three word-hyperbolic groups, Mahan Mj has shown that the inclusion map from H to G extends continuously to a map between the Gromov boundaries of H and G. This boundary map is known as the Cannon-Thurston map. In this context, Mj associates to every point z in the Gromov boundary of Q an ending lamination'' on H which consists of pairs of distinct points in the boundary of H. We prove that for each such z, the quotient of the Gromov boundary of H by the equivalence relation generated by this ending lamination is a dendrite, that is, a tree-like topological space. This result generalizes the work of Kapovich-Lustig and Dowdall-Kapovich-Taylor, who prove that in the case where H is a free group and Q is a convex cocompact purely atoroidal subgroup of Out(FN), one can identify the resultant quotient space with a certain R-tree in the boundary of Culler-Vogtmann's Outer space.

Javier de la Nuez González (U Basque)
Elementary equivalence of graph products of groups
Abstract: We provide a general set of conditions under which the group resulting from a graph product construction interprets both the underlying graph and the collection of groups at its vertices. This implies a form of elementary rigidity for a large class of graph products generalizing the fact that two elementarily equivalent finite graph products of finite groups must be isomorphic. (Joint work with M. Casals-Ruiz, A. Garreta Fontelles and I. Kazachkov)

Boris Kunyavskii (Bar-Ilan University)
Wide simple groups and Lie algebras
Abstract: We say that a group G is wide if its derived subgroup [G,G] contains an element which is not representable as a single commutator of elements of G. Recently it was proven that a finite simple group cannot be wide, thus confirming a conjecture of Ore of 1950's. On the other hand, during the past decades there were discovered several examples of wide infinite simple groups.
In a similar vein, we say that a Lie algebra is wide if its derived algebra [L,L] contains an element which is not representable as a single Lie bracket. A natural question to ask is whether there exist wide simple Lie algebras. We present first examples of such Lie algebras.
This talk is based on a work in progress, joint with Andriy Regeta.

Simon Blackburn (Royal Holloway, U London)
The Walnut Digital Signature Algorithm
Abstract: Walnut is a digital signature algorithm that uses ideas from braid groups. It was first proposed in 2017 by Anshel, Atkins, Goldfeld and Gunnells and was one of the submissions to the high-profile NIST Post-Quantum Cryptography standardisation process. The talk will describe Walnut, some of the attacks that have been mounted on it, and some of the open problems that remain. No knowledge of cryptography will be assumed. Based on joint work with Ward Beullens (KU Leuven).

Unconditionally secure public key transport (with possible errors)
Abstract: We consider a scenario where one party wants to transmit a secret key to another party in the presence of a computationally unbounded (passive) adversary. The legitimate parties succeed with probability close to 1 (although strictly less than 1), while a passive adversary succeeds in correctly recovering the secret key with significantly lower probability, which is independent of the adversary's computational capabilities.
This is joint work with Mariya Bessonov and Dima Grigoriev.

Noam Kolodner (Hebrew U)
Algebraic extensions in free groups and Stallings graphs
Abstract: I will give a counterexample to a conjecture by Miasnikov, Ventura and Weil, that an extension of free groups is algebraic iff the corresponding morphism of Stallings core graphs is onto, for every basis of the ambient group. In the course of the proof I present a partition of the set of homomorphisms between free groups that may be of independent interest.

The simultaneous conjugacy problem in right-angled-artin groups
Abstract: We show that the simultaneous conjugacy problem in a right-angled Artin group can be reduced to finding the centralizer of n given words. In the context of this problem, a solution is known for one general word but not for general n given words. We present a polynomial solution also for this problem.

Ivan Levcovitz (Technion)
Comparing the Roller and B(X) boundaries of CAT(0) cube complexes
Abstract: The Roller boundary is a well-known compactification of a CAT(0) cube complex X. When X admits a cocompact action by a group G and satisfies appropriate hypotheses, Nevo-Sageev show that a subset, B(X), of the Roller boundary is a realization of the Poisson boundary and that the action of G on B(X) is minimal and strongly proximal. After giving a background on these boundaries, I will discuss a characterization of when B(X) is the entire Roller boundary. I will give applications to right-angled Coxeter/Artin groups and to 2-dimensional CAT(0) cube complexes where stronger results hold.

Doron Puder (Tel Aviv)
Group theory at the service of Number theory: the case of Markoff triples
Abstract: The Markoff group of transformations is a group Γ of affine integral morphisms, which is known to act transitively on the set of all positive integer solutions to the equation x2+y2+z2=xyz. The fundamental strong approximation conjecture for the Markoff equation states that for every prime p, the group Γ acts transitively on the set X(p) of nonzero solutions to the same equation over Z/pZ. Recently, Bourgain, Gamburd, and Sarnak proved this conjecture for all primes outside a small exceptional set. We study a group of permutations obtained by the action of Γ on X(p), and show that for most primes, it is the full symmetric or alternating group. We use this result to deduce that Γ acts transitively also on the set of nonzero solutions in a big class of composite moduli. Finally, our result also translates to a parallel in the case r=2 of a well-known theorem of Gilman and Evans regarding "Tr-systems" of PSL(2,p).
This is joint work with Chen Meiri.

Oren Becker (Hebrew U)
Group theoretic stability
Abstract: A finitely generated group G is stable (in permutations) if every approximate action of G on a finite set is close to an actual action. For example, all finitely generated abelian groups are stable (Arzhantseva-Paunescu 2015). The stability of Z2 is equivalent to the statement "almost commuting permutations are near commuting permutations".
I will present a characterization of stability among amenable groups in terms of invariant random subgroups and discuss stability of Kazhdan groups and a quantitative aspect of stability.
Based on joint works with Alex Lubotzky, Andreas Thom and Jonathan Mosheiff.

Curtis Kent (Brigham Young, UT)
Metrics on fundamental groups
Abstract: Many natural groups arise as fundamental groups of spaces without universal covers. Here we will present an approach to endowing fundamental groups of one-dimensional and planar spaces with a metric-like function which allow us to apply techniques from geometric group theory to the study of non-finitely generated fundamental groups. In particular, we will show that the fundamental group of a connected, locally path-connected, planar or one-dimensional space completely determines the homotopy type of the space.

Itay Glazer (Weizmann)
On uniform behavior of families of random walks induced by word maps
Abstract: Given a word, i.e an element w in a free group Fr on a set of r elements, and a finite group G, one can associate a map w:Gr→G referred to as a word map. A word map as above induces a natural probability measure on G, and varying over different finite groups G, we can study the corresponding family of random walks. It turns out that these random walks have uniform behavior over certain interesting families of finite groups. In this talk we will discuss this phenomenon and provide an algebro-geometric interpretation of it. This is a work in progress (joint with Yotam Hendel).

Group equations with automorphisms.
Abstract: We study group equations with occurrences of automorphisms and deal with algebraic sets defined by such type of equations. We proved the criterion when a group G with the given subgroup of its automorphisms A is an equational domain (i.e. any finite union of algebraic sets over G is again algebraic). This result allows us to solve some general problems in universal algebraic geometry and gives the progress in the duscussion of B.Plotkin`s problem for wreath products.

Becca Winarski (Michigan)
Twisted rabbits and invariant trees
Abstract: The twisted rabbit problem is a celebrated problem in complex dynamics. Work of Thurston proves that up to equivalence, there are exactly three branched coverings of the sphere to itself satisfying certain conditions. When one of these branched coverings is modified by a mapping class, a map equivalent to one of the three coverings results. Which one?
After remaining open for 25 years, this problem was solved by Bartholdi-Nekyrashevych using iterated monodromy groups. In joint work with Belk, Lanier, and Margalit, we present an alternate solution using topology and geometric group theory that allows us to solve a more general problem.

Gilbert Levitt (Caen)
On the first order theory of Baumslag-Solitar groups
Abstract: I will discuss various aspects of Baumslag-Solitar groups, in particular of their elementary theory (joint work with Vincent Guirardel)

Alexei Kanel-Belov (Bar Ilan)
Weil algebra and polynomial symplectoeutomorphisms
Abstract: In the papers Y. Tsuchimoto, Endomorphisms of Weyl algebra and p-curvatures, Osaka Journal of Mathematics, vol. 42(2) (2005); A. Kanel-Belov and M. Kontsevich, Automorphisms of Weyl algebras, Lett. Math. Phys. 74 (2005), 181-199; A. Kanel-Belov and M. Kontsevich, The Jacobian Conjecture is stably equivalent to the Dixmier Conjecture, arXiv: math/0512171v2, 2005 was constructed and discoursed homomorphism between (auto)endomorphisms of Weil algebra and polynomial symplectoendo(auto)morphisms. The construction dependant on the choice of infinitely large prime. We prove that for symplectoauthomorphisms in general case and symplectoendomorphisms in deformed case when [xi,\partialj]=ℏδij or {xi,yj}=ℏδij.
My talk concerns recent progress made in the positive resolution of Kontsevich's conjecture, which states that, the procedure utilizes the following essential features. First, the Weyl algebra over an algebraically closed field of characteristic zero may be identified with a subalgebra in a certain reduced direct product (reduction modulo infinite prime) of Weyl algebras in positive characteristic -- a fact that allows one to use the theory of Azumaya algebras and is particularly helpful when eliminating the infinite series. Second, the lifting is performed via a direct homomorphism Aut Wn→Aut Pn which is an isomorphism of the tame subgroups (that such an isomorphism exists is known due to our prior work with Kontsevich) and effectively provides an inverse to it. Finally, the lifted automorphism is the limit (in formal power series topology) of a sequence of lifted tame symplectomorphisms; the fact that any polynomial symplectomorphism has a sequence of tame symplectomorphisms converging to it is our development of the work of D. Anick on approximation and is very recent. In order to make approximation work (this is not trivial at all because the ind-schemes are not reduced), we play with Plank constants and use singularity trick see coming to non-deformed case (it means that plank constant is not small parametr any more) is rather non-trivial and we don't understand how to proceed in endomorphism case. See
[1] and [2] for details.
(joint work with A. Elishev and J.-T. Yu)

Alejandra Garrido (Newcastle, Australia)
Locally compact topological full groups of Cantor set homeomorphisms
Abstract: Much attention has been devoted in recent years to topological full groups of homeomorphisms of a Cantor set, the study of which has yielded new examples of finitely generated infinite simple groups with various properties: amenability (Juschenko-Monod), torsion and intermediate growth (Nekrashevych).
Simple groups play an important role in the emerging theory of totally disconnected locally compact groups, especially the compactly generated ones.
I will report on some joint work with Colin Reid and Dave Robertson, where we study topological full groups (rather, piecewise full groups) that admit a non-discrete locally compact second countable topology. We show that all such groups are abstractly simple and their topology is determined by the group structure. Moreover, taking the piecewise full group of totally disconnected locally compact groups acting 'nicely' on a Cantor set yields many new examples of compactly generated, abstractly simple, totally disconnected locally compact (non-discrete) groups, akin to Neretin's group of almost automorphisms of a regular tree.

Karen Vogtmann (Warwick)
The rational Euler characteristic of Out(Fn)
Abstract: The rational Euler characteristics of many arithmetic groups can be expressed in terms of values of zeta functions, and Harer and Zagier showed that the same is true for surface mapping class groups. For Out(Fn) a generating function for the rational Euler characteristic was given by J. Smillie and myself in 1987 but its asymptotic behavior was not at all clear. I will discuss recent joint work with M. Borinsky that determines the asymptotic behavior and also uncovers a relation with zeta functions.

Aner Shalev (Hebrew U)
Words, probability and representations
Abstract: I will discuss word maps on finite and infinite groups, with emphasis on probabilistic aspects and on free subgroups. I will then focus on recent solutions of some probabilistic Waring problems, obtained in joint work with Larsen and Tiep; these involve new developments in representation theory and character bounds, which are of independent interest. Applications will also be given.

Olga Kharlampovich (Hunter College CUNY)
First order properties of Group Algebras of Free and Limit Groups
Abstract: We will show that geometry and subgroups of a limit group are first-order definable in the theory of its group algebra (in contrast with the situation in a free group). We will also give elementary classification of group algebras of limit groups and prove the undecidability of the Diophantine problem for these algebras. This is based on the joint paper with A. Miasnikov in the Annals of Pure and Applied Logic.

Emmy Noether Lecture by Gregory Lawler, 2019 Wolf Prize laureate
Random Fractals and Critical Phenomena
Abstract: Mathematical models from statistical physics generate random fractal structures at criticality, that is, at values of a parameter at which there is a phase transition. Understanding this rigorously has been one of the focuses of probability theory in the last forty years. I will survey some of the rigorous results focusing on three main models: Brownian motion and exceptional subsets, self-avoiding walk, and loop-erased random walk. An important aspect is the role that the spatial dimension plays in the analysis. I will also discuss open problems and challenges for the future.

Jean Pierre Mutanguha (Arkansas)
Hyperbolic Immersions of Free Groups
Abstract: We proved that the mapping torus of a graph immersion has word-hyperbolic fundamental group if and only if the corresponding endomorphism doesn't generate Baumslag-Solitar subgroups. We will discuss the main ideas behind the proof and the difficulty in generalizing to all graph maps.

Motiejus Valiunas (Southampton)
Acylindrical hyperbolicity of graph products
Abstract: Graph products are a class of groups that interpolate between direct and free products, and include widely studied examples such as the right-angled Artin groups. It is known that most of such groups belong to the class of acylindrically hyperbolic groups, which also includes and shares many properties with (relatively) hyperbolic groups. A recent result suggests a relation between this class and the class of equationally noetherian groups: these are groups for which any system of equations has the same solution set as some finite subsystem.
In this talk I will explicitly describe an action of a graph product on a space quasi-isometric to a tree, which gives an alternative way to prove acylindrical hyperbolicity of graph products. I will discuss some applications, including (but possibly not limited to) a proof that the property of being equationally noetherian is preserved under forming certain graph products.

Soumya Dey (IISER Bhopal, India)
Generalized braid groups and their commutator subgroups
Abstract: The main goal of this talk would be to discuss about various generalizations of the Artin's braid groups, namely the welded braid groups WBn, the twin groups TWn, also known as Grothendieck's cartographical groups, the generalized virtual braid groups GVBn, and the singular braid groups SGn. We shall talk about the structure of the commutator subgroups of these interesting groups. The main tools we have used to prove our results are the Reidemeister-Schreier method and the Tietze transformations. This talk is based on joint works with Dr. Krishnendu Gongopadhyay.

Fabienne Chouraqui (Haifa)
Some approaches to the Herzog-Schonheim conjecture
Abstract: Let G be a group and H1,...,Hs be subgroups of G of indices d1,...,ds respectively. In 1974, M. Herzog and J. Schonheim conjectured that if {Hiαi}i=1i=s, αi ∈ G, is a coset partition of G, then d1,...,ds cannot be distinct. We consider the Herzog-Schonheim conjecture for free groups of finite rank and present some new combinatorial approaches.

Alan Logan (Heriott-Watt U)
Abstract: The Post Correspondence Problem is a decision problem about homomorphisms of free monoids. It was shown to be insoluble by Post in 1946, and has since been much-studied by computer scientists. In a 2014 paper Myasnikov, Nikolaev and Ushakov generalised the problem to groups. In this talk we give positive results for free groups, and explain where the difficulty lies in solving the problem for all free groups.

Classification of torsion-free R-groups
Abstract: All nilpotent torsion-free R-groups will be classified with accuracy up to geometrical equivalence. The corresponding equivalence classes will be described axiomatically with the help of quasi-identities.

Bogdan Stankov (Ecole Normale Superieure)
Limit behaviour of random walks on Schreier graphs
Abstract: The Poisson boundary is a probability space that encodes the limit behaviour of random walks. It is known that a group is amenable if and only if there exists a non-degenerate measure such that the random walk on its Cayley graph has trivial Poisson boundary. When a group acts on a space, the Poisson boundary of the induced walk on the Schreier graph is a quotient of the Poisson boundary of the random walk on the Cayley graph. We discuss results around the non-triviality of the Poisson boundary of the induced walk on the Schreier graph under the hypothesis of a measure with finite first moment. We apply it to Thompson's group F, which extends a result by Kaimanovich about finitely supported measures. We also adapt a similar approach to prove non-triviality of Poisson boundary on subgroups that are not locally solvable of a group H(ℤ) of piecewise projective homeomorphisms. Monod studied the class of groups H(A) of piecewise projective homeomorphisms where A is a subring of the real numbers, and proved that for A≠ℤ, H(A) is non-amenable without free subgroup.

Daniel Woodhouse (Technion)
One-ended hyperbolic groups that are not abstractly co-hopfian
Abstract: I will give the first examples of a one ended hyperbolic group G that contains a finite index subgroup G' that embeds as an infinite index subgroup of G. I will discuss the context for this property and make some conjectures about what the general results may look like.

Chenxi Wu (Rutgers)
Kazhdan's theorem on metric graphs
Abstract: There is an analogy between the study of metric graphs and the study of Riemann surfaces, and a question is to construct uniformization theorem for metric graphs which would require a concept of "hyperbolic metric" on it. With Farbod Shokrieh, we found a graph theoretic analogy of a classical result by Kazhdan on the limit of canonical, or Bergman metric under a tower of normal covers, which indicates that the limiting metric might be such a candidate. I will also discuss generalizations of it to higher dimensional simplicial complex and some further questions.

Stephan Tornier (U Newcastle, Australia)
Abstract: In this TED-like talk we stress the importance of groups acting on trees by situating their theory within all of group theory.

Tengiz Bokelavadze (Akaki Tsereteli State U, Kutaisi, Georgia)
Subgroup lattices and the Geometry of Halls W-power group
Abstract: A study was conducted to demonstrate subgroup lattices and the geometry of Hall's W-power groups. The notion of a discrete W-power group was introduced by Hall who generalized some results of the theory of nilpotent groups to such groups. W-power nilpotent groups played an important role in the general theory of abstract groups, as any finitely generated torsion-free nilpotent group was embedded in some W-power group.

Ji-Young Ham (Chung-Ang U, Seoul)
Golden ratio on nonorientable surfaces of odd genus
Abstract: On each nonorientable surface of odd genus, we give a mapping class which has an invariant subsurface and whose dilatation on this subsurface is the golden ratio.

Christophe Petit (U Birmingham and Oxford)
Rubik's for cryptographers: Babai's conjecture, hash functions and quantum gates
Abstract: Hard mathematical problems are at the core of security arguments in cryptography. In this talk, I will discuss mathematical generalizations of the famous Rubik.s cube puzzle. I will relate them to a conjecture of Babai on the diameter of finite simple groups, to the security of particular cryptographic constructions, to the design of efficient quantum circuits, and more.

Vitali Roman'kov (Omsk State U)
Cryptanalysis via linear algebra and protection against it
Abstract: In the first part of the talk, we discuss an attack, termed a linear decom-position attack, on many of known in literature group-based cryptosystems. This attack gives a polynomial time deterministic algorithm that recovers the secret shared key from the public data in the schemes under consideration. Furthermore, we show that in this case, contrary to the common opinion, the typical computational security assumptions are not very relevant to the security of the schemes, i.e., one can break the schemes without solving the algorithmic problems on which the assumptions are based.
In a number of monographs and papers, we have shown that in many systems of algebraic cryptography, where the platform group G is a subset in a linear space over a finite or infinite field, we can efficiently solve the computational Diffie-Hellman-like problems and hence to compromise the corresponding cryptographic systems. Other and in some points similar approach was established by Tsaban et al.
Also we discuss a {non-linear decomposition} attack that can be applied in many other cases, in particular, when the platform is a polycyclic group. We present two general schemes for which many schemes are specific realizations. One of these two schemes joins schemes based on two-side multiplications, the second scheme joins schemes based on automorphisms. The two mentioned above attacks show vulnerability of these two general schemes.
In the second part of the talk, we introduce a novel method that is resistant against linear algebra attacks. In particular, we propose an improved version of the famous Anshel-Anshel-Goldfeld algebraic cryptographic key-exchange scheme, that is in particular resistant against the Tsaban et al. linear span cryptanalysis. Unlike the original version, that based on the intractability of the simultaneous conjugacy search problem for the platform group, the proposed version is based on much more hard simultaneous membership-conjugacy search problem and needs to solve the membership problem for a subset of the platform group that can be easily and efficiently built as very complicated and without any good structure. A number of other hard problems should be previously solved by any intruder to start solving of the simultaneous membership-conjugacy search problem to obtain the exchanged key. We also show how this new approach can be used to improve many schemes based on the conjugacy search algorithmic problem.

Ben Fine (Fairfield U)
The Axiomatics of Free group Rings
Abstract: In a series of papers, Fine, Gaglione, Rosenberger and Spellman [FGRS 1-2-3] examined the relationship between the universal and elementary theory of a group ring R[G] and the corresponding universal and elementary theory of the associated group G and ring R. These are relative to an appropriate logical language L0, l1, L2 for groups, rings and group rings respectively. Axiom systems T0, T1, T2 for L0, l1, L2 respectively were provided in [FGRS 1]. In [FGRS 1] it was proved that if R[G] is elementarily equivalent to S[H] with respect to L2 then simultaneously the group G is elementarily equivalent to the group H with respect to L0 and the ring R is elementarily equivalent to the ring S with respect to L1. We then let F be a rank 2 free group and Z be the ring of integers. Examining the universal theory of the free group ring the hazy conjecture was made that the universal sentences true in F[G] are precisely the universal sentences true in F modified appropriately for group ring theory and the converse that the universal sentences true in F are the universal sentences true in Z[F] modified appropriately for group theory. In this paper we show this conjecture to be true in terms of axiom systems for Z[F]. Myasnikov and Remeslennikov [MR] have given axiom systems for the universal theory of nonabelian free groups. In this paper we extend this to axiom systems for free group rings and show they are of the same form.
Joint with Anthony Gaglione, Martin Kreuzer, Gerhard Rosenberger and Dennis Spellman.

Thibault Godin (Montpellier & Lorraine)
Mealy automata, groups and growth
Abstract: Mealy automata are powerful tools to generate groups with unusual properties. Since the eighties, they have been used to solve several important group theoretical questions, and are especially ubiquitous in problems related to growth. The growth of a group (namely the function counting the number of elements in balls of growing radii in the Cayley graph) is a geometric way to understand and classify groups, and I will briefly explain how the underlying automaton strucutre gives a powerful leverage to understand it.

Gil Goffer (Weizmann Inst)
The conjugacy problem for almost automorphisms of a tree
Abstract: The group of almost-automorphisms of a regular tree is a beautiful example of a locally compact totally disconnected group. It enjoys surprising properties, together with a rich collection of subgroups of independent interest, such as the Thompson group V. Recently, it was shown to be the first known l.c simple group admitting no lattices.
We present a solution to the conjugacy problem for this group, which uses its unique dynamics when acting on the tree boundary. This is a work in progress, joint with Waltraud Lederle

Volodymyr Nekrashevych (Texas A&M)
Amenable torsion groups
Abstract: The classical methods of constructing groups of Burnside type (infinite finitely generated torsion groups) typically produce non-amenable groups. For example, it is known that all Golod-Shafarevich groups and all known groups of bounded exponent are non-amenable. The only exceptions are some groups generated by automata (e.g., the Grigorchuk group). We will discuss a new class of groups of Burnside type constructed using topological dynamical systems and etale groupoids. A rough idea of the method is "deforming" a locally finite group to produce a finitely generated group, but preserving some of the finiteness condition of the group. This class of groups includes the known examples of amenable groups of Burnside type, but it also includes many more examples (e.g., simple groups and new groups of intermediate growth). Some open problems and directions for further research will be discussed.

Eliyahu Rips / Agata Atkarskaya (Hebrew U / Bar Ilan)
A Group-like Small Cancellation Theory for Rings
Abstract: Our goal is to build a ring that has a good reason to be called a small cancellation ring. In more details we have the following situation.
Let a group $G$ be given by generators and defining relations. It is known that we cannot extract specific information about the structure of $G$ using the defining relations in general case. However, if this defining relations satisfy small cancellation conditions, then we possess a great deal of knowledge about $G$.
Let $kF$ be the group algebra of the free group $F$ over some field $k$. Assume $F$ has a fixed system of generators, then its elements are reduced words in these generators that we call monomials. Let $\mathcal{I}$ be ideal of $kF$ generated by a set of polynomials and let $kF / \mathcal{I}$ be the corresponding quotient algebra. Drawing inspiration from small cancellation conditions on a presentation of a group, we state conditions on these polynomials that enable a combinatorial description of the quotient algebra $kF / \mathcal{I}$, which is similar to a description of small cancellation quotients of the free group. So, the obtained object can be called a group-like small cancellation ring.
We suggest two possible applications of rings defined in the present work. It is known that finitely presented small cancellation groups are hyperbolic. Although the notion of hyperbolic ring does not yet exists, a structure theory developed in this work can be the first step towards a definition of a hyperbolic ring. Also it is known that construction of groups with exotic properties makes an extensive use of small cancellation theory and its generalizations. Generalizations of our approach allows to construct various examples of algebras with exotic properties.
This is a joint work with A. Kanel-Belov, E. Plotkin and E. Rips.

Ruth Charney (Brandeis)
Outer Space for RAAGs
Abstract: Culler and Vogtmann.s Outer Space has played a central role in the study of automorphism groups of free groups. In this talk, I will present some current work with Bregman and Vogtmann on constructing a more general version of Outer Space to study automorphism groups of right-angled Artin groups.

Arman Darbinyan (ENS Paris)
On a problem of Collins about the word and conjugacy problems in groups
Abstract: In early 1970's, Donald Collins asked whether every finitely generated torsion-free group with decidable word problem embeds in a finitely generated group with decidable conjugacy problem. In my talk I will provide a negative answer to this question.

Arie Juhasz (Technion)
A solution of the word problem in even Artin groups
Abstract: An Artin group is called even if the length of each defining relation in the standard presentation has length divisible by 4. This class of Artin groups generalizes the class of right angled Artin groups in which every defining relator has length 4. In this talk we show that f(n)=n6 is an isoperimetric function for even Artin groups. We use small cancellation theory of relative extended presentations.

Last updated: 30 May 2019