Topology & Group Theory Seminar

Vanderbilt University

2019/2020

Links to seminar schedule for previous years:

2010/11,
2011/12,
2012/13,
2014/15,
Spring 2016,
Fall 2016,
Spring 2017,
2017/18,
2018/19

Organizer: Spencer Dowdall

Wednesdays, 4:10–5:00pm in SC 1308 (unless otherwise noted)

** Wednesday, August 28, 2019 **

Kevin Schreve (University of Chicago)

Title: Proper actions versus uniform embeddings

Absract: Whenever a finitely generated group \(G\) acts properly discontinuously by isometries on a metric space \(X\), there is an induced uniform embedding (a Lipschitz and uniformly proper map) \(f \colon G\to X\) given by mapping \(G\) to an orbit. I will talk about some examples of groups which uniformly embed into a contractible n-manifold but do not act on a contractible n-manifold. Kapovich and Kleiner constructed torsion-free hyperbolic groups that embed into \(\mathbb{R}^3\) but only act on \(\mathbb{R}^4\). My main application will be showing that certain k-fold direct products of these groups do not act on \(\mathbb{R}^{3k}\).

Matthew Haulmark (Vanderbilt)

Title: Canonical JSJ for Relatively Hyperbolic Groups

Abstract: JSJ decompositions first appeared in the context of 3-manifolds with the work of Jaco-Shalen and Johannson. Generalizations of JSJ theory to finitely generated groups have been studied in different contexts by many authors (Kropholler, Rips-Sela, Papasoglu-Swenson, Bowditch, Guirardel-Levitt, etc.). Guirardel and Levitt have shown that if G is one ended and hyperbolic relative to a finite collection of finitely generated subgroups, then there is a relative JSJ tree for G over the class of elementary subgroups. In a joint work with Chris Hruska we show that Guirardel and Levitt’s result can be obtained from the topology of the Bowditch boundary. This implies that the relative JSJ tree for G is a “relative” quasi-isometry invariant.

No Seminar

Simon Andre (Vanderbilt)

Title: Hyperbolicity is preserved under elementary equivalence

Abstract: Zlil Sela proved that a finitely generated group that satisfies the same first-order properties as a torsion-free hyperbolic group is torsion-free hyperbolic. I will explain that this result remains true for hyperbolic groups with torsion, as well as for subgroups of hyperbolic groups, and for hyperbolic and cubulable groups.

Rachel Skipper (OSU)

Title: Finiteness Properties for Simple Groups

Abstract: A group is said to be of type $F_n$ if it admits a classifying space with compact \(n\)-skeleton. We will consider the class of Röver-Nekrachevych groups, a class of groups built out of self-similar groups and Higman-Thompson groups, and use them to produce a simple group of type \(F_{n-1}\) but not \(F_n\) for each \(n\). These are the first known examples for \(n\geq 3\). As a consequence, we find the second known infinite family of quasi-isometry classes of finitely presented simple groups.

Jean Pierre Mutanguha (Arkansas)

Title: The irreducibility of monodromy is a mapping torus invariant

Abstract: An immediate corollary of Nielsen-Thurston classification of surface homeomorphisms is that if two surface homeomorphisms f and g have homeomorphic mapping tori, then f is pseudo-Anosov if and only if g is pseudo-Anosov. Using hyperbolization theorem and rigidity results, the hypothesis can be weakened to quasi-isometric mapping tori. We show an analogous result for free group automorphisms: if two free group automorphisms have isomorphic mapping tori, then the first automorphism is fully irreducible and atoroidal if and only if the other is fully irreducible and atoroidal. This answers a question posed by Dowdall-Kapovich-Leininger.

Levi Sledd (Vanderbilt)

Title: Assouad-Nagata dimension of \(C'(1/6)\) groups

Abstract: Asymptotic dimension is a coarse invariant of metric spaces, introduced by Gromov in 1993 as a large-scale analogue of topological dimension. A related concept is that of Assouad-Nagata dimension, a quasi-isometry invariant which is bounded below by asymptotic dimension. Historically, these two invariants have been hard to distinguish among finitely generated groups. In this talk, we show that any finitely generated \(C'(1/6)\) group has Assouad-Nagata dimension at most \(2\). Using this result we show how to construct, for any \(n,k \in \mathbb N\) with \(n \geq 3\), a finitely generated group of asymptotic dimension \(n\) and Assouad-Nagata dimension \(n+k\).

Jesse Peterson (Vanderbilt)

Title: Von Neumann equivalence

Abstract: Two groups are measure equivalent if they have commuting measure-preserving actions on a sigma-finite measure space, such that each group has a finite-measure fundamental domain. We introduce a coarser equivalence, which we call von Neumann equivalence, by allowing the measure space on which the groups act to be non-commutative, i.e., we allow it to be a von Neumann algebra with a semi-finite normal faithful trace. We will show that many "approximation type" properties, (e.g., amenability, property (T)) which are known to be preserved by measure equivalence, are also preserved by von Neumann equivalence. We will also discuss a number of open problems related to this new notion. This is based on joint work with Ishan Ishan and Lauren Ruth.

Daniel Studenmund (Notre Dame)

Title: Algebra and geometry of finite-index subgroups

Abstract: Given an infinite, discrete group G, we will consider the collection C(G) of its finite-index subgroups. First, we study algebraic properties: The abstract commensurator Comm(G) consists of symmetries of C(G), and can detect surprising data about G. We will discuss some known results and pose questions about Comm(F_2). Second, we study geometric properties: C(G) carries a metric space structure that is studied by subgroup growth. We use this to motivate the more general notion of commensurability growth and discuss recent results. This talk includes discussion of work with Khalid Bou-Rabee, Tasho Kaletha, and Rachel Skipper.

Lauren Ruth (Vanderbilt)

Title: The Baum-Connes correspondence for the pure braid group on 4 strands

Abstract: We calculate the left-hand side and the right-hand side of the Baum-Connes correspondence for the pure braid group on 4 strands, each side relying on different techniques. This is joint work with Sara Azzali, Sarah Browne, Maria Paula Gomez Aparicio, and Hang Wang.

Jason Behrstock (CUNY)

Title: Hierarchically hyperbolic groups: an introduction

Abstract: Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, most cubulated groups, and others. In this talk I'll provide an introduction to studying groups and spaces from this point of view, both describing new tools to use to study these groups and applications of those results. This talk will include joint work with Mark Hagen and Alessandro Sisto.

No Seminar (Thanksgiving break)

Genevieve Walsh (Tufts)

Title: Incoherence and Fibering

Abstract: A group is coherent if every finitely generated subgroup is finitely presented, and incoherent otherwise. A group algebraically fibers if it admits a map to the integers with finitely generated kernel. We will discuss the geometry and importance of these notions, and develop techniques to find witnesses to incoherence and algebraic fibers. We apply these techniques to large classes of groups, including many free by free, surface by surface and surface by free groups. This is joint work with Rob Kropholler.

Alexander Olshanskiy (Vanderbilt)

Title: Groups finitely presented in Burnside varieties

Abstract: S.V. Ivanov's problem of 1992 has been solved. For all sufficiently large odd integers \(n\), the following version of Higman's embedding theorem is proved in the variety \({\cal B}_n\) of all groups satisfying the identity \(x^n=1\). A finitely generated group \(G\) from \({\cal B}_n\) has a presentation \(G=\langle A\mid R\rangle\) with a finite set of generators \(A\) and a recursively enumerable set \(R\) of defining relations if and only if it is a subgroup of a group \(H\) finitely presented in the variety \({\cal B}_n\). It follows that there is a 'universal' \(2\)-generated finitely presented in \({\cal B}_n\) group containing isomorphic copies of all finitely presented in \({\cal B}_n\) groups as subgroups.

Denis Osin (Vanderbilt)

Title: Quasi-isometric diversity of marked groups.

Abstract: I will explain how to use basic tools of descriptive set theory to show that a closed set \(\mathcal S\) of marked groups has \(2^{\aleph_0}\) quasi-isometry classes provided every non-empty open subset of \(\mathcal S\) contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains \(2^{\aleph_0}\) quasi-isometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. They can also be used to prove the existence of \(2^{\aleph_0}\) quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties. This talk is based on a joint work with A. Minasyan and S. Witzel.

Michael Ben-Zvi (Tufts)

Title: Connectedness of non-unique CAT(0) boundaries

Abstract: Gromov showed in 1987 that hyperbolic groups have a well defined visual boundary. That is, if a hyperbolic group \(\Gamma\) acts geometrically on spaces \(X_1\) and \(X_2\), then the boundaries of \(X_1\) and \(X_2\) are homeomorphic. He subsequently asked if the same is true for a CAT(0) group. In 2000, Croke and Kleiner answered this question in the negative using the fundamental group of an amalgam of tori. Later, Croke-Kleiner, Mooney, and Wilson each extend this result in different settings. These result all have something in common: the boundaries described are each not path connected. I will outline Croke-Kleiner's original proof and show how to extend the ideas to boundaries which are arbitrarily connected. This is joint work with Robert Kropholler.

John Ratcliffe (Vanderbilt)

Title: Cusp transitivity in hyperbolic 3-manifolds (Joint work with Steven Tschantz)

Abstract: Let \(S\) be a set and \(k\) an integer such that \(1 \leq k \leq |S|\). An action of a group \(G\) on \(S\) is called \(k\)-*transitive* if for every choice of distinct elements \(x_1,\ldots, x_k\) of \(S\) and every choice of distinct targets \(y_1,\ldots,y_k\) in \(S\), there is an element \(g\) of \(G\) such that \(gx_i = y_i\) for each \(i = 1,\ldots, k\). The term *transitive* means 1-transitive, and actions with \(k > 1\) are called *multiply transitive*.
This talk is concerned with cusped hyperbolic 3-manifolds of finite volume whose group of isometries induces a multiply transitive action on the set of cusps of the manifold. Roger Vogeler conjectured that there is a largest \(k\) for which such \(k\)-transitive actions exist, and that for each \(k \geq 3\), there is an upper bound on the possible number of cusps. Our proof of Vogeler's conjecture will be discussed in this talk.

Talia Fernós (University of North Carolina, Greensboro)

Title: Boundaries and CAT(0) Cube Complexes

Abstract: The universe of CAT(0) cube complexes is rich and diverse thanks to the ease by which they can be constructed and the many of natural metrics they admit. As a consequence, there are several associated boundaries, such as the visual boundary and the Roller boundary. In this talk we will discuss some relationships between these boundaries, together with the Furstenberg-Poisson boundary of a "nicely" acting group.

Carolyn Abbott (Columbia University)

Title: Free products and random walks in acylindrically hyperbolic groups

Abstract: The properties of a random walk on a group which acts on a hyperbolic metric space have been well-studied in recent years. In this talk, I will focus on random walks on acylindrically hyperbolic groups, a class of groups which includes mapping class groups, \(\mathrm{Out}(F_n)\), and right-angled Artin and Coxeter groups, among many others. I will discuss how a random element of such a group interacts with fixed subgroups, especially so-called hyperbolically embedded subgroups. In particular, I will discuss when the subgroup generated by a random element and a fixed subgroup is a free product, and I will also describe some of the geometric properties of that free product. This is joint work with Michael Hull.

No Seminar (Spring Break)

Marek Kaluba (Adam Mickiewicz University, Poland)

Title: Computational approach to property (T)

Abstract: Property (T) is a property of compactly generated group which has been introduced by Kazhdan in 1967. The property, expressed in the language of functional analysis is a very strong rigidity property and has far reaching consequences for the actions and the geometry of a group. During the talk I will briefly discuss the the theoretical results of Ozawa which make the computational approach to the Kazhdan property (T) possible. It is known that property (T) is equivalent to positivity of certain operator in the full group \(*\)-algebra. Surprisingly, this positivity is witnessed by the existence of a sum of (hermitian) squares decomposition of the operator in the real *group ring**. This in turn is equivalent to the feasibility of a certain semi-definite optimisation problem, which amounts to a finite computation. I will describe the algorithm encoding the optimisation problem, and how an (imprecise) numerical solution can be turned into a mathematical proof by using the order structure and the topology of cones in group rings. Since (due to its size) the optimisation problem is out of reach of the state-of-the-art solvers we will show how to use the representation theory of finite groups to exploit the symmetry of the optimisation problem to minimize its size. This leads to constructive computer-assisted proof that \(\operatorname{Aut}(F_{5})\) has Kazhdan's property (T).

Michael Hull (University of North Carolina, Greensboro)

Title/Abstract: TBA

No Seminar (Start of summer holiday)