Topology & Group Theory Seminar

Vanderbilt University

2014/2015

Organizer: Mark Sapir

Wednesdays, 4:10pm in SC 1310 (unless otherwise noted)

** Wednesday, August 27, 2014 **

Speaker: Ashot Minasyan (Southampton, UK)

Title: On universal right angled Artin groups.

Abstract: A right angled Artin group (RAAG), also called a graph group or a partially commutative group, is a group which has a finite presentation where the only permitted defining relators are commutators of the generators. These groups and their subgroups play an important role in Geometric Group Theory, especially in view of the recent groundbreaking results of Haglund, Wise, Agol, and others, showing that many groups possess finite index subgroups that embed into RAAGs.

In their recent work on limit groups over right angled Artin groups, Casals-Ruiz and Kazachkov asked whether for every natural number n there exists a single "universal"
RAAG, A_n, containing all n-generated subgroups of RAAGs. Motivated by this question, I will discuss several results showing that "universal" (in various contexts) RAAGs generally do not exist.

**Wednesday, September 3, 2014 **

Speaker: Gili Golan (Bar Ilan, Israel)

Title: Tarski numbers of group actions

Abstract: The Tarski number of an action of a group G on a set X is the minimal number of pieces in a paradoxical decomposition of it. For any k > 3 we construct a faithful transitive group action with Tarski number k. Since every k<4 is not a Tarski number, this provides a complete characterization of Tarski numbers of group actions. Using similar techniques we construct a group action of a free group F with Tarski number 6 such that the Tarski numbers of restrictions of this action to finite index subgroups of F are arbitrarily large.

** Wednesday, September 10, 2014 **

Speaker: Jesse Peterson (Vanderbilt)

Title: A remark about spectral radii

Abstract: We show that if a unitary representation of a discrete group does not contain almost invariant vectors, then there exist finite subsets whose corresponding Markov operators have spectral radii tending to zero. This generalizes a result of Andreas Thom who considered this question for the left-regular representation of non-amenable groups.

** Wednesday, September 17, 2014 **

Speaker: Mike Mihalik (Vanderbilt)

Title: The Fundamental Group at Infinity for a Finitely Presented Group

Abstract:
If a finitely presented group satisfies a certain asymptotic
condition called semistability at infinity, then the fundamental group
at an end of that group is independent of base
ray converging to that end
(in analogy with a space being path connected so that
fundamental group is independent of base point).
The following are long standing open (and associated) questions:
Question 1: Are all finitely presented groups semistable at infinity?
Question 2: Is H^{2}(G, ZG) free abelian for all finitely presented
groups G?
We begin this talk with motivation, history, examples and classical
results associated with these questions. We end the talk with a proof of the following:
Theorem. If a finitely presented group G contains an infinite, finitely generated sub-commensurated subgroup of infinite index, then G is semistable at infinity and H^{2}(G,ZG) is free abelian.
This result generalizes many of the classical results
on semistability. If H is a subgroup of a group G then H is commensurated in G if for all g in G, the intersection of gHg^{-1} and H has finite index in both. So commensurated is weaker than normal.

** Wednesday, September 24, 2014 **

Speaker Voughan Jones (Vanderbilt)

Title: Some unitary representations of the Thompson groups F and T

Abstract: In a "naive" attempt to create algebraic quantum field theories on the circle, we obtain
a family of unitary representations of Thompson's groups T and F for any subfactor.
In the simplest case the coefficients of the representations are polynomial invariants
of links and the question arises of just what links the Thompson group produces.

** Wednesday, October 1, 2014 **

Speaker: Yago Antolin Pichel (Vanderbilt)

Title: Commuting degree for infinite groups

Abstract: There is a classical result saying that, in a finite group, the probability that two elements commute is never between 5/8 and 1 (i.e. if it is greater than 5/8 the group is abelian).
In this talk we present a generalization of this result for infinite finitely generated groups.
The main result is the following The main one is the following: "A polynomially growing group G
has positive commuting degree if and only if it is virtually abelian". This is a Joint work with Enric Ventura and Armando Martino.

** Wednesday, October 8, 2014 **

Speaker: Kun Wang (Vanderbilt)

Title: A structure theorem for Farrells twisted Nil-groups

Abstract: Farrell Nil-groups are generalizations of Bass Nil-groups to the twisted case.
They mainly play role in (1) The twisted version of the Fundamental theorem
of algebraic K-Theory (2) Algebraic K-theory of group rings of virtually cyclic groups (3) as the obstruction to reduce the family of virtually cyclic groups used in the Farrell-Jones conjecture to the family of finite groups. These groups are quite mysterious. Farrell proved in 1977 that Bass Nil-groups are either trivial or infinitely generated in lower dimensions. Recently, we extended Farrells result to the twisted case in all dimensions. We indeed obtained a structure theorem for an important class of twisted Nil-groups. This is a joint work with Jean Lafont and Stratos Prassidis.

** Wednesday, October 15, 2014 **

Speaker: Andrew Sale (Vanderbilt)

Title: Some right-angled Artin groups whose automorphism groups do not have Property (T).

Abstract:
Grunewald and Lubotzky described a family of representations of finite index subgroups of the automorphism group of a non-abelian free group, Aut(F_{n}). In particular, they showed that the image of these representations are arithmetic groups, and furthermore one particular subfamily of these representations have image SL(n-1,Z). A consequence of this is that Aut(F_{3}) does not have Kazhdan's Property (T). I will describe how these representations can be generalised to automorphism groups of right-angled Artin groups and hence describe a condition sufficient to imply when they do not have Property (T)

** Wednesday, October 29, 2014 **

Speaker: Denis Osin (Vanderbilt)

Title: Highly transitive actions and acylindrical hyperbolicity.

Abstract: I will discuss my work in progress with Michael Hull. We prove that every countable acylindrically hyperbolic group admits a highly transitive action with finite kernel. This theorem uniformly generalizes many previously known results and allows us to answer a question of Garion and Glassner on the existence of highly transitive actions of mapping class groups. It also implies that, in certain geometric and algebraic settings, transitivity degree of a faithful permutation representation of an infinite groups can only take two values, namely 1 and infinity. Further, for any countable group G admitting a highly transitive faithful action, we prove the following dichotomy: Either G contains a normal subgroup isomorphic to the infinite alternating group or G resembles a free product in a certain precise sense. We discuss applications of this theorem to the study of universal theory and mixed identities of acylindrically hyperbolic groups.

** Wednesday, November 5, 2014 **

Speaker: Mike Mihalik (Vanderbilt)

Title: TBA

Abstract: TBA