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 H2(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 H2(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(Fn). 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(F3) 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: Finitely Generated Groups that are Simply Connected at Infinity and Subcommensurability.

Abstract: We introduce the notion of simple connectivity at infinity for finitely generated groups (in analogy with that for finitely presented groups). We use this new concept, along with the main (semistability) theorem presented in the 9/17/14 seminar to prove the following theorem:

Theorem. If a finitely generated group G contains a 1-ended, finitely presented sub-commensurated subgroup of infinite index, then G is simply connected at infinity. If additionally, G is finitely presented then H2(G,ZG) is trivial.

We cannot prove the (more simple) finitely presented version of this theorem without the notion of a simply connected at infinity finitely generated group. (Sub)normal subgroups are (sub)commensurated and so this theorem is valid if subcommensurated is replaced by subnormal (solving a problem we have been interested in since the early 1980s). In 1981 B. Jackson proved the following:

Theorem. (B. Jackson) If a finitely presented group G contains a 1-ended, infinite, finitely presented and normal subgroup of infinite index, then G is simply connected at infinity.

Wednesday, November 12, 2014

Speaker: Yago Antolin Pichel (Vanderbilt)

Title: Graph products, walls and the Haagerup property

Abstract: A graph product is a construction that generalizes free and direct products. I will explain how normal forms of graph product define a natural wall structure. We will use this walls to promote wall spaces from the vertex groups to the whole group. As a consequence we will show that the Haagerup property is preserved under graph products or that the graph product of groups acting geometrically on CAT(0) cube complexes act on on a CAT(0) cube complex.

Wednesday, November 19, 2014

Jennifer Hom (Columbia University)

Title: An infinite rank summand of topologically slice knots

Abstract: The knot concordance group consists of knots in the 3-sphere, modulo the equivalence relation of smooth concordance. The group operation is induced by connected sum, and the identity element is generated by slice knots. We will consider the subgroup T generated by topologically slice knots. Endo showed that T contains an infinite rank subgroup, and Livingston and Manolescu-Owens showed that T contains a rank 3 summand. We will show that in fact T contains an infinite rank summand. The proof relies on knot Floer homology.

Wednesday, December 3, 2014

Jessica Purcell (BYU)

Title: Geometrically maximal knots

Abstract: The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. This motivates several questions, such as, for which knots is the ratio very near the upper bound? For fixed crossing number, which knots have largest volume or determinant? We show that many families of alternating knots and links simultaneously maximize both ratios, and investigate related questions for these families. This is joint work with Abhijit Champanerkar and Ilya Kofman.