Topology & Group Theory Seminar

Vanderbilt University

2017/2018

Organizer: Mark Sapir

Wednesdays, 4:10pm in SC 1308 (unless otherwise noted)

** Wednesday, August 30, 2017 **

Mike Mihalik (Vanderbilt)

Title: Semistability of relatively hyperbolic groups.

Abstract:
Suppose *G * is a 1-ended finitely presented group that is hyperbolic relative to ** P ** a finite collection of 1-ended finitely generated subgroups. Our main theorem states that if

∂ (G, ** P **) has no cut point, then *G * has semistable fundamental group at ∞. Under mild conditions on *G* and the members of ** P** the 1-ended hypotheses and the no cut point condition can be eliminated to obtain the same semistability conclusion.

** Wednesday, September 6, 2017 **

Matthew Haulmark (Vanderbilt)

Title: A classification theorem for 1-dimensional boundaries of groups with isolated flats.

Abstract: In 2000 Kapovich and Kleiner proved that if G is a one-ended hyperbolic group that does not split over a two-ended subgroup, then the boundary of G is either a Menger curve, a Sierpinski carpet, or a circle. Kim Ruane observed that there were no known non-hyperbolic examples of groups with Menger curve boundary, and asked if there was a CAT(0) generalization of Kapovich and Kleiners theorem. As boundaries of CAT(0) groups are in general not locally connected, there is no hope of such a generalization for general CAT(0) groups. However, a version of Kapovich and Kleiners theorem may hold for certain classes of CAT(0) groups. In this talk I will discuss a generalization of the Kapovich-Kleiner theorem for CAT(0) groups with isolated flats, and provide an example of a non-hyperbolic CAT(0) group with Menger curve boundary.

**Wednesday, September 13, 2017**

Arman Darbinyan (Vanderbilt)

Title: Word and conjugacy problems in finitely generated groups

Abstract: We will discuss some new results about the relationship between word and conjugacy problems in finitely generated groups.
In particular, we will discuss a method which allows us to construct: (1) finitely generated (solvable or finitely presented) torsion-free groups with
decidable word problem and such that they cannot be embedded into groups with decidable conjugacy problem;
(2) finitely generated groups with decidable semi-conjugacy problem and undecidable conjugacy problem.
Both (1) and (2) answer questions which were known as open questions.

** Wednesday, September 27, 2017**

Caglar Uyanik (Vanderbilt)

Title: Dynamics and geometry of free group automorphisms

Abstract: I will talk about the long standing analogy between the mapping class group of a hyperbolic surface and the outer automorphism group of a free group. Particular emphasis will be on the dynamics of individual elements and applications of these results to structural theorems about subgroups of these groups.

** Wednesday, October 4, 2017 **

Mladen Bestvina (University of Utah)

Title: Boundary amenability of *Out(F _{n})*

Abstract: I will discuss boundary amenability and how to prove it for
basic groups for most of the hour. The main interest in boundary
amenability is that it implies the Novikov conjecture in manifold
theory. I will then outline the main ideas in the proof of boundary
amenability of *Out(F _{n})*. This is joint work with Vincent Guirardel and
Camille Horbez.

Vaughan Jones (Vanderbilt)

Title: Coefficients of certain unitary representations of Thompson groups.

Abstract: There is a general construction, essentially due to Ore, of a group of quotients
of certain rather special categories. If the category is planar rooted binary forests (under stacking),
the group of quotients is Thompsons group *F*. Actions of these groups of fractions can be constructed
whenever there is a functor from the underlying category to another category that admits direct limits.
I will briefly review these constructions and show how to construct many actions of F, including
unitary representations. Irreducibility is then a major question. As a first attempt to tackle this problem
I will focus on the *coefficients* of these unitary representations. Calculation of coefficients
led to a construction of knots and links but we will see that it also leads to the study of iteration of
dynamical systems which in the simplest cases are rational functions on *ℂ P ^{1}*. We will
show some pictures of Julia and Fatou sets of these maps.

Valeriano Aiello (Vanderbilt)

Title: TBA

Abstract: TBA

** Wednesday, Novbember 8, 2017 **

Matt Clay (University of Arkansas)

Title: TBA

Abstract: TBA

** Wednesday, December 6, 2017**

Hung Cong Tran (University of Georgia)

Title: TBA

Abstract: TBA

**Wednesday, February 14, 2018**

Anthony Genevois (Aix-Marseille University, France)

Title: TBA

Abstract: TBA