Shanks Workshop: Geometric Methods in Group Theory

Schedule (pdf)

All talks will be at Buttrick Hall 101.

Saturday, April 23:

Sunday, April 24:

Abstracts of talks

Mladen Bestvina (Utah)

Title: On the Farrell-Jones conjecture for mapping class groups

Abstract.
I will try to describe what the Farrell-Jones conjecture is about, and how one
goes about proving it. Then I will try to outline a proof of FJC for mapping
class groups, which is work in progress, joint with Arthur Bartels.

Michel Hull (UIC)

Title: Homomorphisms to acylindrically hyperbolic groups.

Abstract. We will discuss the study of homomorphisms to an acylindrically hyperbolic group, or equivalently the study of systems of equations in an acylindrically hyperbolic group.
In particular, we will give a criteria for an acylindrically hyperbolic group G to be equationally noetherian, which means that G satisfies a group-theoretic version of the Hilbert basis theorem.
As an application, we will show that a group which is hyperbolic relative to equationally noetherian subgroups is equationally noetherian. We will also discuss some potential future applications to 3-manifold groups.

Sergei V. Ivanov (UIUC)

Title: The bounded word problem and the precise word problem for presentations of groups.

Abstract. We discuss the bounded word problem and the
precise word problem for groups given by generators and defining relations. For
example, for every finitely presented group, the bounded word problem is in NP
and the precise word problem is in PSPACE. It is proved that, for certain
finite presentations of groups, which include Baumslag-Solitar one-relator
groups and free products of cyclic groups, the bounded word problem and the
precise word problem can be solved in polylogarithmic space. As consequences of
developed techniques that can be described as calculus of brackets, we obtain
polylogarithmic space bounds for the computational complexity for the diagram
problem for free groups, for the width problem for elements of free groups, and
for computation of the area defined by polygonal singular closed curves in the
plane. Since a polylogarithmic space bound automatically implies a
quasipolynomial time bound, we also obtain quasipolynomial time bounds for
these problems.

Olga Karlampovich (CUNY)

Title: Tarski-type questions for group rings

Abstract. We consider some fundamental model-theoretic
questions that can be asked about a given algebraic structure (a group, a ring,
etc.), or a class of structures, to understand its principal algebraic and logical
properties. These Tarski type questions include: elementary classification and
decidability of the first-order theory.

We describe solutions to Tarski’s problems in the class of group algebras of free
groups. We will show that unlike free groups, two groups algebras of free
groups over infinite fields are elementarily equivalent if and only if the
groups are isomorphic and the fields are equivalent in the weak second order
logic. We will also show that for any field, the theory of a group algebra of a
torsion free hyperbolic group is undecidable and for a field of zero
characteristic even the diophantine problem is undecidable. (These are joint
results with A. Miasnikov)

Ashot Minasyan (Southampton)

Title: An exotic group action on an R-tree.

Abstract. I will discuss a construction of a finitely
generated group L and an R-tree T such that 1) L acts on T non-trivially and
with finite arc stabilizers; 2) L has Serre’s property (FA), i.e., any action
of L on a simplicial tree has a global fixed vertex. To construct the group L we
employ small cancellation theory over hyperbolic groups, developed by
Ol’shanskii. The tree T is obtained as a strong limit (in the sense of Gillet
and Shalen) of simplicial Bass-Serre trees corresponding to certain amalgamated
free products.

Piotr Przytycki (McGill)

Title: Rips complex for relatively hyperbolic groups

Abstract. We will describe a Rips complex, a thickening of
the Cayley graph of a relatively hyperbolic group G, with a graph-theoretic
property called dismantlability. This guarantees fixed-point properties and
implies that the Rips complex is a classifying space for G (with respect to
appropriate family). This is joint work with Eduardo Martinez-Pedroza.

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