Topology & Group Theory Seminar
Vanderbilt University
Spring 2022

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 2019/20 Fall 2020 Spring 2021 Fall 2021

Organizers: Spencer Dowdall & Denis Osin

Time: Wednesdays, 4:10–5:00pm US Central Time (unless otherwise indicated)

Location: SC 1312 for in-person talks (alternately via Zoom for remote talks)

Format: The seminar will typically meet in-person. However, the seminar will occasionally meet Online via Zoom Meeting as the situation merits; such as for outside speakers who are not able to travel to Vanderbilt. Please email one of the organizers for the Zoom Meeting invitation and join link.

Mailing List: If you would like to be added the seminar mailing list and receive regular announcments, please contact an organizer.

Wednesday, February 16, 2022

Yuri Bakhturin (Memorial University, Canada)

Title: Group Gradings and Actions of Pointed Hopf Algebras

Abstract: Pointed Hopf algebras are a wide class of Hopf algebras, including group algebras and enveloping algebras of Lie algebras. In this talk, based on a recent work with Susan Montgomery, we study actions of pointed Hopf algebras on simple algebras. These actions are known to be inner, as in the case of Skolem - Noether theorem. We try to give an explicit description, whenever possible, and consider Taft algebras, their Drinfeld doubles and some quantum groups.

Wednesday, February 23, 2022
Online Zoom Talk – contact organizers for Zoom Meeting invitation link.

Yvon Verberne (Georgia Tech)

Title: Automorphisms of the fine curve graph

Abstract: The fine curve graph of a surface was introduced by Bowden, Hensel and Webb. It is defined as the simplicial complex where vertices are essential simple closed curves in the surface and the edges are pairs of disjoint curves. We show that the group of automorphisms of the fine curve graph is isomorphic to the group of homeomorphisms of the surface, which shows that the fine curve graph is a combinatorial tool for studying the group of homeomorphisms of a surface. This work is joint with Adele Long, Dan Margalit, Anna Pham, and Claudia Yao.

Wednesday, March 2, 2022

Christopher J. Leininger (Rice)

Title: Purely pseudo-Anosov subgroups of fibered 3-manifold groups

Abstract: Farb and Mosher, together with work of Hamenstädt, proved that Gromov hyperbolicity for surface group extensions is entirely encoded by algebraic and geometry properties of the monodromy into the mapping class group. They were thus able to give a purely geometric formulation for Gromov’s Coarse Hyperbolization Question for the class of surface group extensions: Given a finitely generated, purely pseudo-Anosov (free) subgroup of the mapping class group, is it convex cocompact? In this talk, I will explain joint work with Jacob Russell in which we answer the question affirmatively for subgroups of fibered 3-manifold groups, completing a program for such groups begun in earlier work with Kent and Schleimer, and continued in work with Dowdall and Kent.

Wednesday, March 16, 2022

Alex Margolis (Vanderbilt)

Title: A cornucopia of simple lattices in products of trees

Abstract: The class of groups acting properly and cocompactly on the direct product of two locally finite trees is much richer than one might initially expect, as demonstrated by work of Wise and Burger-Mozes. Most notably, Burger-Mozes exhibited the first known examples of finitely presented infinite simple torsion-free groups within this class of groups.

In this talk, I will discuss a random model for investigating lattices in the product of trees. With high probability, the groups in this random model are just-infinite. I will also discuss a counting result on the number of virtually simple groups in this class, giving an abundance of virtually simple finitely presented groups. This is joint work with Nir Lazarovich and Ivan Levcovitz.

Wednesday, March 30, 2022

Radhika Gupta (Temple)

Title: Orientable maps and polynomial invariants of free-by-cyclic groups

Abstract: Given a graph map from a graph to itself, we can associate two numbers to it: geometric stretch factor and homological stretch factor. I will define a notion of orientability for graph maps and use it to characterize when the two numbers are equal. The notion of orientability can be upgraded to certain automorphisms of free groups as well. A (fully irreducible) automorphism f of a free group determines a free-by-cyclic group to which we can associate two polynomial invariants: the McMullen polynomial and the Alexander polynomial. These polynomials determine the stretch factor and homological stretch factor of f. We will see how orientability helps us to relate these two polynomials. This is joint work with Spencer Dowdall and Samuel Taylor.

Wednesday, April 6, 2022

Samuel Shepherd (Vanderbilt)

Title: Semistability of cubulated groups

Abstract: I will discuss my theorem that cubulated groups are semistable at infinity, together with background on these two concepts. I will also present a result about modifying the cubulation of a group to achieve certain geometric features, which is needed to prove the semistability theorem.

Wednesday, April 13, 2022
Online Zoom Talk – contact organizers for Zoom Meeting invitation link.

Tarik Aougab (Haverford)

Title: Detecting covers, simple closed curves, and Sunada's construction

Abstract: Given a pair of finite degree (not necessarily regular) covers (p,X),(q,Y) of a finite type surface S, we show that the covers are equivalent if and only if the following holds: for any closed curve gamma on S, some power of gamma admits an embedded lift to X if and only if some power of gamma admits an embedded lift to Y. We apply this to study the well-known construction of Sunada which yields pairs of hyperbolic surfaces (X,Y) that are not isometric but that have the same unmarked length spectrum. In particular we show that the length-preserving bijection from closed geodesics on X to those on Y arising from the Sunada construction never sends simple closed geodesics to simple closed geodesics. We also show that length-isospectral surfaces arising from several of the most well-known manifestations of the construction are not simple length isospectral. Even more, we construct length-isospectral hyperbolic surfaces so that for each finite n, the set of lengths corresponding to closed geodesics with at most n self intersections disagree. This represents joint work with Maxie Lahn, Marissa Loving, and Nicholas Miller.

Wednesday, April 20, 2020
No seminar -- due to Math Award Ceremony
(Note that Ionut Chifan's seminar scheduled for April 21 has been cancelled)

Wednesday, April 27, 2022

Srivatsav Kunnawalkam Elayavalli (Vanderbilt)

Title: Sofic approximations of groups

Abstract: I will present a result of Ben Hayes and myself, where we show that any initially subamenable group is either amenable or there exists two sofic embeddings that are not conjugate by any automorphism. This generalizes some work of Elek-Szabo and answers a question of Paunescu. Importantly, the proof uses von Neumann algebras in a fundamental sense.