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)

Schedule:
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 ErdosRenyi 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 FalgasRavry.
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
IAautomorphisms or the Torelli subgroup of
A, respectively. I will discuss various finiteness properties of subgroups containing
G_{N}, the Nth term of the lower central series of
G, for sufficiently small
N. In particular, I will explain why
(1) If
n≥4N1, then any subgroup of
G containing
G_{N} (e.g. the Nth term of the Johnson filtration) is finitely generated
(2) If
n≥8N3, then any finite index subgroup of A containing
G_{N} 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
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 quasiregular 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 UrbanaChampaign)
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.
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.
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.
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 subexponential growth. Some explicit examples will be
presented.
Towards a grouplike small cancellation theory for rings
This is a joint work with Agata Atkarskaya, Alexei KanelBelov 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 = x^{a} y x^{(a+1)} y ... x^{(b1)} 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.
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.
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(
ℝ^{n}), of crystallographic groups Γ acting isometrically on
ℝ^{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
ndimensional Euclidean space
ℝ^{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(
ℝ^{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
π_{1}(M) contains a free group ?
We will explain ideas and methods, recent and old results related to the above problems.
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 Kfinite, where
K stands for any of the five Green's relations, if every finitely generated semigroup from
V has only finitely many
Kclasses. 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 quasiconvex subgroups of automatic groups
The notion of Stallings graphs can be extended from finitely generated subgroups of free groups, to quasiconvex 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 quasiconvex subgroups) such as the generalized membership problem, or the conjugacy and the almost malnormality problems for quasiconvex 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.