Abstracts
Emmanuel Breuillard
Title: Random character varieties.
Abstract: We study the representation and character varieties of random finitely presented groups with values in a complex semisimple Lie group.We compute the dimension and number of irreducible components of the character variety of a random group. For example we show that for all but exponentially few pairs of words (w_1,w_2) of length at most n, all homomorphic images of the finitely presented group <a,b| w_1=w_2=1> in GL(d,C) have virtually solvable image, or that random one-relator groups <a,b | w> have many rigid Zariski-dense representations. The proofs are conditional on GRH and use expander graphs of finite simple groups of Lie type as a key ingredient.
Martin Bridson
Title:Virtual surjection to k-tuples, binary subgroups, and finiteness properties.
Abstract: The Virtual Surjection to Pairs (VSP) Theorem states that if a subgroup S of a direct product of finitely presented groups projects to a subgroup of finite index in each pair of factors, then S is finitely presented. If S is a full subdirect product of free or surface groups, the converse also holds. I shall discuss analogous results concerning higher finiteness properties and projections to k-tuples of factors, and I shall exploit these in an elementary construction that provides a new source of groups with varying finiteness properties -- the binary subgroups of direct products.
Corentin Bodart
Title: Rational cross-sections, bounded generation and orders on groups
A rational cross-section is a regular language of unique representatives for the elements of a group. Rational cross-sections are linked to many subjects in geometric group theory: they are part of the definition of automatic (or even autostackable) groups, byproducts of finite (or even regular) complete rewriting systems, and a key tool to compute growth series of groups. In particular, many groups are known to admit rational cross-sections. In contrast very few groups without rational cross-section were known. In this talk I'll present some new examples of groups without rational cross-sections, specifically wreath products similar to $\mathbb{Z}\wr F_2$, the second Houghton group, and even a finitely presented HNN extension of Grigorchuk's group. Our arguments rely on connections with (regular) left-invariant orders on groups and bounded generation.
Elena Bunina
Title: Universal and AE theories of linear groups
We present and discuss recent results on universal and AE-equivalence of linear groups.
We show that two linear groups G_n(R) and G_m(S), where
G\in {GL, SL} and either n,m\ge 2, R,S be arbitrary fields or n,m\ge 3, $R,S$ be local rings with 1/2, are universally equivalent if and only if n=m and R and S are universally equivalent (Maltsev-style results). Also we show the same results for the symplectic linear groups Sp_{2n}(R) and Sp_{2m}(S), where n,m\ge 1 and R,S are fields with 1/2.
Then we show that there are some difficulties with similar results for projective linear groups, but the same type of theorems takes place for AE-equivalence: two linear groups or projective linear groups (GL, PGL, SL, PSL, Sp, PSp) G_n(R) and G_m(S) of the same types over fields R and S are AE-equivalent if and only if n=m and R and S are AE-equivalent.
Then we discuss AE-definability of linear groups. While classes of any type of linear groups are not universally definable, some of them are AE-definable. We will show the recent results here.
Joint with Galina Kaleeva and Alexey Lazarev.
Cornelia Drutu
Title: Connections between hyperbolic geometry and median geometry
Abstract: In this talk I shall explain how groups endowed with various forms of hyperbolic geometry, from lattices in rank one simple groups to acylindrically hyperbolic groups, present various degrees of compatibility with the median geometry. This is joint work with Indira Chatterji, and with John Mackay.
David Fisher
Title: Greenberg-Shalom and Lyndon-Ullman
Abstract: Recently Yehuda Shalom has revived and expanded an old
speculation
of Leon Greenberg. If G is a simple Lie group, might it be the case
that any discrete
subgroup \Gamma in G with dense commensurator is in fact a lattice in G?
And therefore an arithmetic lattice by a theorem of Margulis. In joint
work with Brody, Mj and van Limbeek, we show that if this speculation
is correct, it has myriad implications. Including resolving an old question
of Lyndon and Ullman, recently made a conjecture by Kim and Koberda,
about groups generated by non-commuting parabolic in SL(2,R).
I will discuss some of these connections and also some partial
results on the Greenberg-Shalom hypothesis that are joint work
with Mj and van Limbeek.
Damien Gaboriau
Title: On the homology torsion growth for some classical groups such as SL_d(Z), Artin groups and mapping class groups
Abstract:
The growth of the sequence of Betti numbers is quite well understood when considering a suitable sequence of finite sheeted covers of a manifold or of finite index subgroups of a countable group.
We are interested in other homological invariants, like the growth of the mod p Betti numbers and the growth of the torsion of the homology. We produce new vanishing results on the growth of torsion homologies in higher degrees for SL(d,Z), mapping class groups, Out(Wn) and Artin groups. As a central tool, we introduce a quantitative homotopical method that constructs "small" classifying spaces for finite index subgroups, while controlling at the same time the complexity of the homotopies. Our method easily applies to free abelian groups and then extends recursively to a wide class of residually finite groups.
I will present the basic objects and some of the ideas.
This is mainly joint work with Miklos Abert, Nicolas Bergeron and Mikolaj Fraczyk.
Anthony Genevois
Title: Are right-angled Artin groups really right-angled angled?
Abstract: The main goal of the talk will be to motivate the extension of reflection groups to rotation groups, a rotation being thought of as fixing pointwise a separating subset and permuting freely-transitively the (possibly infinitely many) components. This approach offers a common point of view on Coxeter groups and graph products of groups such as right-angled Artin groups (which appear as right-angled rotation groups). Geometrically, this suggests the introduction of a new family of graphs, called plurilith graphs, which includes (quasi-)median graphs and Cayley graphs of Coxeter groups.
Susan Hermiller
Title: Formal conjugacy growth for graph products
Abstract: The conjugacy growth series of a finitely generated group measures the growth of conjugacy classes, in analogy with the standard growth series that measures the growth of elements of the group. In contrast, though, conjugacy growth series are rarely rational, and even for free groups with standard generating sets, the series are transcendental and their formulas are rather complicated. In this talk I will discuss several results on conjugacy growth and languages in graph products, including a recursive formula for computing the conjugacy growth series of a graph product in terms of the conjugacy growth and standard growth series of subgraph products. In the special case of right-angled Artin groups I will also discuss a another formula for the conjugacy growth series based on a natural language of conjugacy representatives. This is joint work with Laura Ciobanu and Valentin Mercier.
Camille Horbez
Title: Right-angled Artin groups and measure equivalence
Abstract: Measure equivalence, introduced by Gromov, is a measure-theoretic analogue of quasi-isometry. It is tightly related to orbit equivalence of group actions on standard probability spaces. I will report on joint work with Jingyin Huang, where we establish rigidity/flexibility/classification results for right-angled Artin groups from this viewpoint.
Andrei Jaikin-Zapirain
Title: Profinite properties of residually finite groups.
Abstract: A group property of a residually finite group is profinite if it can be detected by looking at the finite images of the group. In my talk I will present some progress on the next two questions.
1. Is being residually-$p$ a profinite property?
2. Let $G$ be a residually finite group and $H$ a subgroup of $G$. Can we detect that $H$ is a free factor of $G$ in finite quotients of $G$?
The second part of my talk is based on a joint work with Alejandra Garrido.
Nir Lazarovich
Title: Ascending chains of free subgroups in 3-manifold groups
Abstract: In joint work with Edgar A. Bering IV, we show that every ascending chain of free subgroups of constant rank in a 3-manifold group must stabilize. In this talk we will discuss the proof of this result and see how it brings the geometrization, hyperbolic geometry and Bass-Serre theory into play.
Yash Lodha
Title: A tale of two algebraic notions: left orderability and local indicability.
Abstract: Whether a countable group admits a faithful action by orientation preserving homeomorphisms on R admits a striking algebraic characterisation. Such an action exists iff the group is left orderable, i.e. it admits a total order which is invariant under group (left) multiplication. A group is said to be locally indicable if every finitely generated subgroup admits a homomorphism onto the integers. This property emerged in Higman's study of units in group rings, and has found applications in the study of various long standing open problems in group theory and topology. The goal of the talk is to understand to what extent these properties differ: and the exotic examples that lurk around the boundary.
Alex Lubotzky (Weizmann Inst. )
Title: Good locally testable codes
Abstract: An error-correcting code is locally testable (LTC) if there is a random tester that reads only a small number of bits of a given word and decides whether the word is in the code, or at least close to it. A long-standing problem asks if there exists such a code that also satisfies the golden standards of coding theory: constant rate and constant distance. Unlike the classical situation in coding theory, random codes are not LTC, so this problem is a challenge of a new kind. We construct such codes based on what we call (Ramanujan) Left/Right Cayley square complexes. These are 2-dimensional versions of the expander codes constructed by Sipser and Spielman (1996).
The main result and lecture will be self-contained. But we hope also to explain how the seminal work of Howard Garland ( 1972) on the cohomology of quotients of the Bruhat-Tits buildings of p-adic Lie group has led to this construction ( even though it is not used at the end). Based on joint work with I. Dinur, S. Evra, R. Livne, and S. Mozes.
Nicolàs Matte Bon
Title:On the geometry of graphs of actions of solvable groups
Abstract: Given a group G, we are interested in the geometric properties shared by all graphs of faithful actions of G. in particular in their volume growth. It is well known that many finitely generated groups G admit faithful actions whose Schreier graph is much smaller than the Cayley graph of the group itself, for instance there are groups of exponential growth which admit faithful actions whose graph grows linearly. We aim at studying obstructions to the existence of such actions of small growth. To formalise this, given a function f(n), we say that a group G has Schreier growth gap f(n) if every faithful G-set has growth at least f(n). I will discuss results establishing Schreier growth gaps in the setting of finitely generated solvable groups. This is joint work with Adrien Le Boudec.
Alexei Miasnikov
Title: First-order classification, general algebraic schemes, and non-standard
groups
In this talk I will discuss a new notion of an algebraic group scheme and
the related class of “new” algebraic groups (which, of course, contains
the classical ones). This leads to some interesting results on the first-
order classification problems and sheds new light on the nature
of the first-order rigidity and quasi finite axiomatization. In another
direction I will touch on non-standard models of groups (aka non-
standard analysis), in particular, non-standard models of the finitely
generated ones with decidable word problems, and explain how they
naturally appear as non-standard Z-points of the general algebraic
groups.
Vladimir Nekrashevych
Title: Growth estimates for simple groups of intermediate growth
This is joint work with Tianyi Zheng and Laurent Bartholdi. I will discuss a new approach to obtaining growth estimates of the form $\exp(n^a)$ for simple groups of intermediate growth. The usual method of "strong contracting" developed for groups acting on rooted trees does not work in this setting. We replace it with the study of complexity of the action of the group and show exponential decay of the number of "traverses".
Eugene Plotkin
Bounded generation for Chevalley and Kac-Moody groups and their logical properties
We discuss recent results on bounded elementary generation and bounded commutator width for Chevalley groups over Dedekind rings of arithmetic type in positive characteristic. In particular, Chevalley groups of rank greater than 1 over polynomial rings and Chevalley groups of arbitrary rank over Laurent polynomial rings (in both cases the coefficients are taken from a finite field) are boundedly elementarily generated. We establish rather plausible explicit bounds. We discuss applications to Kac-Moody groups and various model theoretic consequences. If time permits, we focus attention on logical questions for ind-algebraic groups. Finally, we plan to state conjectures which look quite tempting. Joint work with B.Kunyavskii, N.Vavilov
Aner Shalev
Title: Groups, characteristic covering numbers, and tensor product growth of representations.
Abstract: I will present two recent joint works with Michael Larsen and Pham Tiep.
The first shows that short products of words (in disjoint variables) which are not an identity of
any (non-abelian) finite simple group are surjective on ALL finite simple groups. This is carried
out in a more general framework of characteristic collections of groups, and may be regarded
as a generalization of work by Guralnick, Liebeck, O'Brien, Tiep and me (2018), which in turn
extends the Odd Order Theorem.
The second part of the talk will focus on representations of finite simple groups, exploring related
growth phenomena. It is motivated by the theory of approximate subgroups, which establishes
3-step growth phenomena for simple groups of Lie type of bounded rank. We replace products of
subsets by tensor products of representations, and establish stronger 2-step growth results also
in unbounded rank.
Rizos Sklinos
Title: First-order sentences in random groups
Abstract: Gromov in his seminal paper introducing hyperbolic groups claimed that a “typical” finitely presented group is hyperbolic. His statement can be made rigorous in various natural ways and was proved correct. In a different line of thought, Tarski asked whether all non abelian free groups share the same first-order theory (in the language of groups). This question proved very hard to tackle and only after more than 50 years Kharlampovich-Myasnikov and Sela answered the question positively. Combining the two, J. Knight asked whether a first-order sentence is true in a typical group if and only if it is true in a no abelian free group. In joint work with O. Kharlampovich we answer the question positively for universal sentences.
Katrin Tent
Title: Sharply 2-transitive groups and the Burnside problem
Abstract: Until recently all known sharply 2-transitive groups were of the form
$K\rtimes K^*$ for some (near-)field K. The first sharply 2-transitive
groups without abelian normal subgroups were constructed in joint work
with Segev and Rips. I will present recent progress on estabilishing the
existence of such groups with additional properties and will explain the
connection to the Burnside problem.
Giulio Tiozzo
Title: The sublinearly Morse boundary
Abstract: Several notions of boundaries of groups have been developed, with advantages and disadvantages.
One of the most famous constructions is the Gromov boundary, which captures asymptotic classes
of quasi-geodesics, but can only be defined for hyperbolic groups.
Another construction is the Morse boundary, which is invariant under quasi-isometry, but “too small”,
in the sense that it has measure zero with respect to most natural measures.
We build an analogue of the Gromov boundary for any proper geodesic metric space that
is both invariant under quasi-isometry and has full measure with respect to hitting measures for random walks.
As an application, when G is the mapping class group of a finite type surface, or a relatively hyperbolic group, then with minimal assumptions the Poisson boundary of G can be realized on the sublinearly Morse boundary of G equipped with the word metric associated to any finite generating set.
Joint with Yulan Qing and Kasra Rafi.
Matthew Tointon
Transience of random walks on vertex-transitive graphs via growth and isoperimetry in groups.
A random walk on a graph is called recurrent if it eventually returns to its starting point with probability 1, and transient otherwise. Varopoulos famously showed that the simple random walk on an infinite vertex-transitive graph G is recurrent if and only if G is quasi-isometric to the standard Cayley graph of Z or Z^2. In this talk I will describe how recent quantitative work on growth and isoperimetry of groups, joint with Romain Tessera, allows us to prove a quantitative, finitary version of Varopoulos's theorem. Amongst other things this proves that there is a gap at 1 for the return probability of the simple random walk on a vertex-transitive graph, i.e. an absolute constant eps>0 such that if the probability that the walk eventually returns to its starting point is at least 1-eps then the walk is recurrent. It also verifies (indeed, strengthens) an analogue of Varopoulos's result for finite graphs conjectured by Benjamini and Kozma in 2002.
Alina Vdovina
Title: Classifying new higher-dimensional analogues of the Thompson groups by K-theory of C*-algebras.
Abstract: We present explicit constructions of infinite families of CW-complexes of arbitrary dimension with buildings as the universal covers. These complexes give rise to new families of C*-algebras, classifiable by their K-theory.
The underlying building structure allows explicit computation of the K-theory. We will present new higher-dimensional generalizations of the Thompson groups, which are usually difficult to distinguish, but the K-theory of C*-algebras gives new invariants to recognize non-isomorphic groups.
We will also discuss new directions of generalizations to higher dimensions of the work of Vaughan Jones and his collaborators on connections of the Thomson's group and Theoretical Physics.
Karen Vogtmann
Title: Gamma-complexes
The right-angled Artin group A_Gamma based on a graph Gamma acts properly on a contractible space of “marked Gamma-complexes" called O_Gamma. For Gamma a complete graph on n vertices O_Gamma is the symmetric space SL(n,R)/O(n) and a Gamma-complex is a flat n-torus. For Gamma a discrete graph O_Gamma is Outer space and a Gamma-complex is a finite graph with fundamental group F_n. We describe Gamma-complexes for general Gamma, and explain how to recognize whether a CAT(0) cube complex with fundamental group A_Gamma is indeed a Gamma-complex. This is joint work with Corey Bregman and Ruth Charney.
Yaroslav Vorobets
Title: Topological full group of the Morse subshift.
Abstract: The famous Morse sequence is obtained by starting with 0 and repeatedly applying the substitution rule 0-->01, 1-->10. The sequence gives rise to a dynamical system, a substitution subshift, which is a minimal homeomorphism of a Cantor set.
The topological full group of this dynamical system consists of all homeomorphisms which are piecewise powers of the subshift. It is a countable, finitely generated group. The talk is concerned with group-theoretical properties of this group. The main result is that it has a subgroup isomorphic to one of the Grigorchuk groups.
The Morse subshift has a continuous factor which is itself a substitution subshift, generated by the substitution 0-->01, 1-->00. It turns out that the Grigorchuk group naturally embeds into the topological full group of the factor while the latter naturally embeds into the topological full group of the Morse subshift.
The talk is based on joint work with Rostislav Grigorchuk.
Pascal Weil
About the finitely generated subgroups of the modular group
I will explain how one can count, exactly and asymptotically, the subgroups of size n of the modular group PSL_2(Z) --- that is, the subgroups whose Stallings graph has n vertices ---, and how this can be used to efficiently generate these subgroups uniformly at random. Any such subgroup is isomorphic to a free product of k copies of Z_2, \ell copies of Z_3 and a free group of rank r, and we compute the expected value of these three parameters (namely n^{1/2} for k and n^{1/3} for \ell). We also show a large deviations theorem: the isomorphism type of a random subgroup is highly concentrated around its expected value.
We also briefly discuss another method to study the same subgroups, this time counting and randomly generating subgroups of a given size and isomorphism type. This second method relies on a rewriting system for Stallings graphs with interesting probabilistic properties. Finally, we ask whether this rewriting system has a geometric interpretation on the surfaces associated with the subgroups of the modular group.
This is joint work with Frederique Bassino and Cyril Nicaud
Armin Weiss
Equation satisfiability for finite solvable groups
Over twenty years ago, Goldmann and Russell initiated the study of the complexity of the equation satisfiability problem (PolSat) and the NUDFA program satisfiability problem (ProgSat) in finite groups.They showed that these problems are decidable in polynomial time for nilpotent groups while they are NP-complete for non-solvable groups.However, for a long time the case of solvable but non-nilpotent groups remained wide open -- in a long sequence of papers only the case of p-by-abelian groups could be shown to be in polynomial time. In 2020 Idziak, Kawałek, Krzaczkowski and myself succeeded to show that in groups of Fitting length at least three, PolSat cannot be solved in polynomial time under the condition that the exponential time hypothesis (ETH) holds. In this talk I will explain this result and also provide some details on very recent work considering PolSat for groups of Fitting length two (ie groups which have a nilpotent normal subgroup with a nilpotent quotient). Moreover, I will explain the related problems ProgSat and ListPolSat for which, under ETH and the so-called constant degree hypothesis, we can get a complete classification in which cases they are in P.