## 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 University)

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 University)

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 University)

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 University)

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 University)

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 University)

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 University)

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 University)

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

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.

Friday, January 16, 2015 (same room and time)

Laura Ciobanu (University of Neuchatel, Switzerland)

Title: Equations in groups

Two natural questions in algorithmic group theory are: is it decidable whether an equation whose coefficients are elements of a given group has at least one solution in that group? And if an equation has solutions, how can we best describe them? The talk will start with a survey on this topic, and will conclude with a language theoretic characterization of the solutions of equations in free groups (joint with M. Elder and V. Diekert) and results concerning the asymptotic behavior of satisfiable homogeneous equations in surface groups (with Y. Antoln and N. Viles).

Wednesday, January 21, 2025

Mark Sapir (Vanderbilt University)

Title: On Jones' subgroup of R. Thompson group F

Abstract: Recently Vaughan Jones showed that the R. Thompson group F encodes in a natural way all knots and links in R3, and a certain subgroup of F encodes all oriented knots and links. We answer several questions of Jones about his subgroup. In particular we prove that the subgroup is generated by x0x1, x1x2, x2x3 (where xi,i in N are the standard generators of F) and is isomorphic to F3, the analog of F where all slopes are powers of 3 and break points are 3-adic rationals. We also show that the subgroup coincides with its commensurator. Hence the linearization of the corresponding permutational representation of F is irreducible. Finally we show how to replace 3 in the above results by an arbitrary n, and to construct a series of irreducible representations of F defined in a similar way. This is a joint work with Gili Golan.

Wednesday, February 4, 2015

Ben Hayes (Vanderbilt University)

Title: Metric Mean Dimension for Algebraic Actions of Sofic Groups

Abstract: Mean dimension and metric mean dimension are dynamical invariants of an action of a group on a compact metrizable space. They were defined for the case when the group is amenable by Lindenstrauss and Weiss, and has been extended to the sofic case by Li. Metric mean dimension can be thought of as a dynamically version of dimension, and is an analogue of entropy for large spaces. For example, the metric mean dimension of a Bernoulli shift is the dimension of the base. In this work, we are concerned with metric mean dimension in the case of actions of a group G by automorphisms of a compact, metrizable abelian group X. We relate metric mean dimension of this action the von Neumann rank of the dual of X as a Z(G)-module. This may be viewed as part of the recent phenomenon of relating the L2-invariants of a Z(G) module A to dynamical properties of the action of G on the Pontryagin dual of A. No knowledge of sofic groups, mean dimension, or von Neumann algebras will be assumed.

Wednesday, February 11, 2015

Alessandro Sisto (ITH, Zurich, Switzerland)

Title: Deviation estimates for random walks and acylindrically hyperbolic groups

Abstract: We will consider a class of groups that includes non-elementary (relatively) hyperbolic groups, mapping class groups, many cubulated groups and C'(1/6) small cancellation groups. Their common feature is to admit an acylindrical action on some Gromov-hyperbolic space and a collection of quasi-geodesics "compatible" with such action. As it turns out, random walks (generated by measures with exponential tail) on such groups tend to stay close to geodesics in the Cayley graph. More precisely, the probability that a given point on a random path is further away than L from a geodesic connecting the endpoints of the path decays exponentially fast in L. This kind of estimate has applications to the rate of escape of random walks (local Lipschitz continuity in the measure) and its variance (linear upper bound in the length). Joint work with Pierre Mathieu.

Wednesday, March 25, 2015

Jason Behrstock (CUNY, Graduate Center and Lehman College)

Title: Coxeter groups, divergence, and random graphs

Abstract: We will present new results on random graphs which are motivated by ideas in geometric group theory. These result, in turn, have applications to Coxeter groups which will also be discussed. Some of this talk will be on joint work with Hagen and Sisto; other parts are joint work with Hagen and Susse.

Wednesday, April 1, 2015

Elizabeth Fink (ENS, Paris, France)

Title: Morse geodesics in torsion groups

Abstract: A geodesic in a metric space is Morse, if quasi-geodesics connecting points on it stay uniformly close. An element is Morse if the embedding of the cyclic subgroup generated by it is a Morse geodesic. In previously known cases it was true that when a group has Morse geodesics, then it also has Morse elements. By studying asymptotic cones, we will exhibit Morse geodesics in infinite torsion groups which are direct limits of hyperbolic groups. On the contrary, it will be shown that there also exist non-Morse geodesics in the same groups, which do not even contain arbitrarily large powers. I will also discuss related properties and possible consequences.

Wednesday, April 8, 2015

Denis Osin (Vanderbilt University)

Title: Acylindrical hyperbolicity of groups with positive first L2-Betti number

Abstract: The main purpose of my talk will be to discuss the following question: Is every finitely presented groups with positive first L2-Betti number acylindrically hyperbolic? I will explain the rationale behind this question and discuss some partial results. In particular, I will show that the answer is positive for residually finite groups.
Wednesday, September 2, 2015

Alexander Olshanskii (Vanderbilt University)

Title: Subnormal subgroups in free groups, their growth and co-growth

Abstract: We (1) compare subnormal closures of finite sets in a free group $F$; (2) obtain the limit for the series of subnormal closures of a single element in $F$; (3) prove that the exponential growth rate (e.g.r.) $\lim_{n\to \infty}\sqrt[n]{g_H(n)}$, where $g_H(n)$ is the growth function of a subgroup $H$ with respect to a finite free basis of $F$, exists for any subgroup $H$ of the free group $F$; (4) give sharp estimates from below for the e.g.r. of subnormal subgroups in free groups; and (5) find the co-growth of the subnormal closures of free generators.

Wednesday, September 9, 2015

Jesse Peterson (Vanderbilt University)

Title: Connes' character rigidity conjecture for lattices in higher rank groups

Abstract: We show that lattices in higher rank center-free simple Lie groups are operator algebraic superrigid, i.e., any unitary representation of the lattice which generates a II_1 factor extends to a representation of its group von Neumann algebra. This generalizes results of Margulis and Stuck-Zimmer, and answers in the affirmative a conjecture of Connes.

Wednesday, September 23, 2015

Yago Antolin Pichel (Vanderbilt University)

Title: Formal conjugacy growth and hyperbolicity.

Abstract: Rivin conjectured that the formal conjugacy growth series of an hyperbolic group is rational if and only if the group is virtually cyclic. In this talk, I will present a proof of Rivin's conjecture and supporting evidence for the analogous statement for acylindrically hyperbolic groups. This is a joint work with Laura Ciobanu.

Wednesday, September 30, 2015

Boris Lishak (University of Toronto)

Title: Balanced finite presentations of the trivial group and geometry of four-dimensional manifolds.

Abstract: We will construct a sequence of finite presentations of the trivial group with just two generators and two relators that can be transformed to the trivial presentation only by means of an enormously large number of Tietze transformations. The proof of the lower bound involves developing a small cancellation theory over "fake" HNN-extensions. We will explain several applications of such results to 4-dimensional geometry. For example, one can prove the existence of non-trivial "thick" 4-knots in the 5-dimensional Euclidean space. One can also construct trivial 2-knots in the 4-sphere that are very difficult to untie. (These geometric applications are a joint work with A. Nabutovsky.)

Wednesday, October 7, 2015

Mark Sapir (Vanderbilt University)

Title: On subgroups of the R. Thompson group F.

Abstract: This is a joint work with Gili Golan. We answer a question of Savchuk, giving examples of maximal subgroups of F which do not fix a point, show how to use a 2-dimensional analog of the Stallings foldings to prove that a subgroup of F does not coincide with F, and show that F is quasi-residually finite.

Wednesday, October 21, 2015

Sam Taylor (Yale University)

Title: Hyperbolic bundles: From 3-manifolds fibering over the circle to hyperbolic extensions of free groups

Abstract: As part of his celebrated hyperbolization theorem, Thurston proved that a fibered 3-manifold M admits a hyperbolic structure if and only if M is the mapping torus of a psuedo-Anosov homeomorphism. Thinking in terms of the induced splitting of the fundamental group of M, Thurstons theorem suggests the possibility of hyperbolizing more general group extensions that live beyond the world of 3-manifold topology. In this talk I will discuss recent work developing the theory of hyperbolic extensions of free groups and highlight the surprising similarities these findings share with Thurston's original theory. In addition to surveying the geometry of these hyperbolic extensions, I will also present some new results which quantitatively describe how topological and combinatorial data is transferred between various fibrations of the 3-manifold M. This talk includes joint work with S. Dowdall, Y. Minsky, and G. Tiozzo.

Wednesday, November 4, 2015

Bryan Jacobson (Vanderbilt University)

Title: Algebraic subgroups of acylindrically hyperbolic groups

Abstract: The Zariski topology on a group G is generated by taking the collection of solution sets to group equations to be a subbasis for the closed sets. A subset of G is called algebraic if it is Zariski-closed. We will discuss a structure theorem for non-elementary algebraic subgroups of acylindrically hyperbolic groups and provide a couple of basic applications.

Wednesday, November 11, 2015

Carolyn Abbot (University of Wisconsin-Madison)

Title:Not all acylindrically hyperbolic groups have universal acylindrical actions

Abstract: The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(F_n) for n > 1. In such a group, a generalized loxodromic element is one that is loxodromic for some acylindrical action of the group on a hyperbolic space. Osin asks whether every finitely generated acylindrically hyperbolic group has an acylindrical action on a hyperbolic space for which all generalized loxodromic elements are loxodromic. In this talk, I will answer this question in the negative, using Dunwoodys example of an inaccessible group as a counterexample.

Wednesday, November 18, 2015

Gili Golan (Vanderbilt University)

Title: On the core of subgroups of Thompson group F

Abstract: The core of a subgroup of Thompson group F is defined using a 2-dimensional analog of Stallings' core graphs. We discuss properties of the core of a finitely generated subgroup of F. As applications, we present an algorithm for deciding whether a finitely generated subgroup of F acts transitively on the set D of finite binary fractions. We show that solvable subgroups of F do not act transitively on D. We also prove that for any rational number x the stabilizer of x in Thompson group F is finitely generated. This is joint work with Prof. Mark Sapir.

Wednesday, December 2, 2015

Rares Rasdeaconu (Vanderbilt University)

Title: Rational curves on real K3 surfaces

Abstract: The simply connected complex surfaces with vanishing first Chern class, also known as K3 surfaces, are discussed from the perspective of their enumerative geometry. Do such surfaces contain rational curves, i.e. holomorphically immersed 2-spheres? If yes, how many? Moreover, if the K3 surface admits an anti-holomorphic involution, how many of these curves are invariant? I will try to give some answers to such questions based on recent joint work with V. Kharlamov.