## Topology & Group Theory Seminar Vanderbilt University 2018/2019

Links to seminar schedule for previous years:
2010/11, 2011/12, 2012/13, 2014/15, Spring 2016, Fall 2016, Spring 2017, 2017/18

Organizer: Spencer Dowdall

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

Wednesday, September 5, 2018

Spencer Dowdall (Vanderbilt)

Title: Abstract commensurations of big mapping class groups

Abstract: It is a classic result of Ivanov that the mapping class group of a finite-type surface is equal to its own automorphism group. Relatedly, it is well-known that non-homeomorphic surfaces cannot have isomorphic mapping class groups. In the setting of "big mapping class groups" of infinite-type surfaces, the situation is more complicated due to the fact that the sheer enormity and variety of behavior prevents group elements from having canonical descriptions in terms of normal forms. This talk will present work with Juliette Bavard and Kasra Rafi overcoming these difficulties and extending the above results to big mapping class groups. In particular, we show that any isomorphism between big mapping class groups is induced by a homeomorphism of the surfaces and that each big mapping class group is equal to its abstract commensurator.

Wednesday, September 12, 2018

Marissa Loving (UIUC)

Title: Least dilatation of pure surface braids

Abstract: The n-stranded pure surface braid group of a genus g surface can be described as the subgroup of the pure mapping class group of a surface of genus g with n-punctures which becomes trivial on the closed surface. I am interested in the least dilatation of pseudo-Anosov pure surface braids. For the n=1 case, upper and lower bounds on the least dilatation were proved by Dowdall and Aougab—Taylor, respectively. In this talk, I will describe the upper and lower bounds I have proved as a function of g and n.

Wednesday, September 19, 2018

Jesse Peterson (Vanderbilt)

Title: Properly proximal groups and their von Neumann algebras.

Abstract: We will introduce a class of groups, which we call properly proximal, which includes all nonelementary hyperbolic groups, all nonelementary bi-exact groups, all convergence groups, all lattices in semisimple Lie groups, and is closed under commensurability and taking direct products, but excludes all amenable and even all inner-amenable groups. We will then discuss rigidity results for von Neumann algebras associated to measure-preserving actions of these groups.

Wednesday, September 26, 2018

Jonathan Campbell (Vanderbilt)

Title: Homotopy Theory, Fixed Point Theory and the Cyclotomic Trace

Abstract: Fixed point theory has been the motivation for many of the most celebrated results of 20th century mathematics: the Lefschetz fixed point theorem, the Atiyah-Singer index theorem, and the development of etale cohomology. In this talk I'll describe work, joint with Kate Ponto, that relates classical fixed point theory to an important homotopy theoretic invariant called algebraic K-theory. The relationship seems to clarify both domains, and readily suggests generalizations that relate to dynamical zeta functions. The link turns on careful considerations of the bicategorical structure of THH. Prerequisites: An appetite for (or, lacking that, a tolerance of) category theory. I will try to define and motivate most objects.

Wednesday, October 3, 2018

Mike Mihalik (Vanderbilt)

Title: Relatively hyperbolic groups with free abelian second cohomology

Abstract: H. Hopf conjectured (probably in the 1940's) that if $$G$$ is a finitely presented group then $$H^2(G;\mathbb{Z}G)$$ is free abelian. While many classes of groups are known to satisfy this conjecture (including all word hyperbolic groups), the conjecture remains open today. For a group $$G$$ we consider a condition on $$H_1^{\infty}(G)$$, the first homology of the end of $$G$$ that is equivalent to $$H^2(G;\mathbb{Z})$$ being free. Suppose $$G$$ is a 1-ended finitely presented group that is hyperbolic relative to $$\mathcal P$$ a finite collection of 1-ended finitely presented proper subgroups of $$G$$. Our main theorem states that if the boundary $$\partial (G,\mathcal{P})$$ is locally connected and the second cohomology group $$H^2(P,\mathbb ZP)$$ is free abelian for each $$P\in \mathcal{P}$$, then $$H^2(G,\mathbb{Z}G)$$ is free abelian. When $$G$$ is 1-ended it is conjectured that $$\partial (G,\mathcal{P})$$ is always locally connected. When $$G$$ and each member of $$\mathcal{P}$$ is 1-ended and $$\partial (G,\mathcal{P})$$ is locally connected, we prove that the "Cusped Space" for this pair has semistable fundamental group at $$\infty$$. This provides a starting point in our proof of the main theorem.

Wednesday, October 10, 2018

James Farre (Utah)

Title: TBA

Abstract: TBA

Wednesday, October 24, 2018

Alexander Olshanskiy (Vanderbilt)

Title: TBA

Abstract: TBA

Wednesday, November 7, 2018

Justin Lanier (Georgia Tech)

Title: TBA

Abstract: TBA