Conference on geometric and combinatorial methods in group theory

In honor of Mark Sapir's 60th birthday

16 - 18 May 2017 University of Illinois at Urbana Champaign


Useful locations:

Registration: Altgeld Hall, in front of room 314
Talks: Altgeld Hall, room 245
Poster session: Altgeld Hall, room 314.
Other coffee breaks: Altgeld, room 321.
Banquet: Colonial room, Illini Union (opposite Altgeld Hall)


Tue. 5/16 Wed. 5/17 Thu. 5/18
Registration and refreshments
8:00am - 9:00am
Coffee and pastries : 8:30am - 9:00am
Rostislav Grigorchuk
9:00am - 10:00am
Eliyahu Rips
9:00am - 10:00am
Mikhail Volkov
9:00am - 10:00am
Coffee break
Tatiana Smirnova-Nagnibeda
10:30am - 11:30am
Sergei Ivanov
10:30am - 11:30am
Jason Behrstock
10:30am - 11:30am
11:30am - 1:00pm
Volodymyr Nekrashevych
1:00pm - 2:00pm
Pascal Weil
1:00pm - 2:00pm
Mikhail Ershov
1:00pm - 2:00pm
Coffee break
Poster session and refreshments 2:00pm - 3:00pm Coffee break
Russell Lyons
2:30pm - 3:30pm
Martin Kassabov
2:30pm - 3:30pm
Efim Zelmanov
3:00pm - 4:00pm
Gregory Soifer
3:45pm - 4:45pm
Hosted by Paul Schupp

Jason Behrstock (CUNY)

Random graphs and applications to Coxeter groups

The divergence function provides a way to measure aspects of the large scale geometry of a group, by quantifying how quickly pairs of geodesic rays separate from each other. Questions about the study of divergence functions were initially raised by Gromov and have received significant attention in the last several years. We will provide an introduction to this invariant and survey results about divergence in finitely generated groups. We will also discuss our recent proof of a threshold theorem for divergence in random groups, which we proved by establishing an Erdos-Renyi style threshold theorem in random graphs. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen, Susse, and Falgas-Ravry.

Mikhail Ershov (University of Virginia)

On finiteness properties of the Johnson filtrations

Let A denote either the automorphism group of a free group of rank n or the mapping class group of an orientable surface of genus n with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of A, respectively. I will discuss various finiteness properties of subgroups containing GN, the Nth term of the lower central series of G, for sufficiently small N. In particular, I will explain why

(1) If n≥4N-1, then any subgroup of G containing GN (e.g. the Nth term of the Johnson filtration) is finitely generated

(2) If n≥8N-3, then any finite index subgroup of A containing GN has finite abelianization.

The talk will be based on a joint work with Sue He and a joint work with Tom Church and Andrew Putman

Rostislav Grigorchuk (Texas A&M)

On unitary representations of groups acting on measure spaces with the application to branch and weakly branch groups.

I will begin with the discussion of Koopman, groupoid and quasi-regular unitary representations that can be naturally associated with a non singular action of a group on a measure space. The matter of the weak containment and the weak equivalence of them will be the main point of the introduction.

Then I will switch to the actions of weakly branch groups on the boundary of rooted tree and show that under the condition of subexponential boundedness one gets uncountably many pairwise non equivalent but weakly equivalent unitary representations belonging to the above types of representations. At the end I will mention totally non free actions and factor representations. The talk will be based on the results from the three joint papers of the speaker with Artem Dudko.

Sergei Ivanov (University of Illinois at Urbana-Champaign)

On some versions of the word problem for presentations of groups

We look at the bounded word problem and the precise word problem for groups given by generators and defining relations and discuss related results. These problems are quantitative versions of the classical word problem.

Martin Kassabov (Cornell University)

Tricks with amenable groups

I will discuss a construction of a large family of finitely generated groups based on the limits of marked groups. As a result we can construct finitely generated groups with uncountably many different spectral radii. This is joint work with I. Pak.

Russell Lyons (Indiana University)

Spectral Embedding Bounds Random Walk Eigenvalues and Return Probabilities

We give a lower bound for the eigenvalues of the Laplacian of finite Cayley graphs, and the spectral measure of infinite Cayley graphs, in terms of a lower bound for the volume growth. These in turn yield upper bounds on the return probabilities of random walks. This is joint work with Shayan Oveis Gharan.

Volodymyr Nekrashevych (Texas A&M University)

Simple torsion groups of intermediate growth

I will describe a new class of infinite finitely generated torsion groups. They are constructed by "fragmenting" an action of the dihedral group on a Cantor set. Under additional conditions on the action and on the fragmentation, one can obtain groups that are simple and of sub-exponential growth. Some explicit examples will be presented.

Eliyahu Rips (Hebrew University)

Towards a group-like small cancellation theory for rings

This is a joint work with Agata Atkarskaya, Alexei Kanel-Belov and Eugen Plotkin.
In group theory, certain combinatorial conditions on the relations, known as small cancellation conditions, provide a great deal of information about the resulting group. With the goal to be able to state similar conditions in the case of rings, we work out a special particular case. Namely, let A be the group ring of a free group F over a field with two elements. Let w be a primitive cyclically reduced word in F. Let x, y be some free generators of F. We take a word v = xa y x(a+1) y ... x(b-1) y, where a and b are integers with a much bigger than the length of w and b much bigger than a. Let I be the ideal of A generated by the element v(1+w), and let B = A/I be the quotient ring. We give a combinatorial description of B similar to that of small cancellation groups.

Tatiana Smirnova-Nagnibeda (University of Geneva)

Spinal groups revisited

We will discuss some old and new results about spinal groups – a class of groups acting on rooted trees, introduced by Bartholdi and Sunic in 2001 as a generalization of Grigorchuk’s family of groups of intermediate growth.

Gregory Soifer (Bar Ilan University)

Properly discontinuous groups of affine transformations. Mathematical developments arising from Hilbert 18th problem.

The study of affine crystallographic groups has a long history which goes back to Hilbert's 18th problem. More precisely Hilbert (essentially) asked if there is only a finite number, up to conjugacy in Aff(n), of crystallographic groups Γ acting isometrically on n. In a series of papers Bieberbach showed that this was so. The key result is the following famous theorem of Bieberbach. A crystallographic group Γ acting isometrically on the n--dimensional Euclidean space n contains a subgroup of finite index consisting of translations. In particular, such a group Γ is virtually abelian, i.e. Γ contains an abelian subgroup of finite index. In 1964 Auslander proposed the following conjecture.

The Auslander Conjecture. Every crystallographic subgroup Γ of Aff(n) is virtually solvable, i.e. contains a solvable subgroup of finite index. In 1977 J. Milnor stated the following question:

Question. Does there exist a complete affinely flat manifold M such that π1(M) contains a free group ?

We will explain ideas and methods, recent and old results related to the above problems.

Mikhail Volkov (Ural Federal University)

Local finiteness for Green relations in semigroup varieties

The lattice of varieties of semigroups is studied with respect to the following concepts: a variety V is said to be locally K-finite, where K stands for any of the five Green's relations, if every finitely generated semigroup from V has only finitely many K-classes. The talk is based on a joint work with Pedro Silva and Filipa Soares.

Pascal Weil (LaBRI, CNRS and Université de Bordeaux)

Algorithmic problems for quasi-convex subgroups of automatic groups

The notion of Stallings graphs can be extended from finitely generated subgroups of free groups, to quasi-convex subgroups of automatic groups (Kapovich, 1996; Kharlampovich, Miasnikov, Weil, 2016). Computing these graphs is not usually as quick as in free groups but it can be done algorithmically. Stallings graphs then offer a unified approach to solve many algorithmic problems (restricted to quasi-convex subgroups) such as the generalized membership problem, or the conjugacy and the almost malnormality problems for quasi-convex subgroups of hyperbolic groups. This is joint work with Olga Kharlampovich (CUNY) and Alexei Miasnikov (Stevens Institute of Technology)

Efim Zelmanov (University of California San Diego)

Matrix Wreath Products.

We will review a new construction of matrix wreath product of algebras that is analogous to wreath product of groups. To prove its usefulness we will discuss applications to embedding theorems and growth functions of algebras.