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.

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.

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.

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.

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.

James Farre (Utah)

Title: Infinite volume and bounded cohomology

Abstract: To a hyperbolic 3-manifold \(M\), we associate the class in cohomology that computes the volume of geodesic tetrahedra in \(M\). We will be interested in the setting that \(M\) has infinite volume, so this cohomology class is necessarily zero. To circumvent this shortcoming, we introduce bounded cohomology. To each hyperbolic structure on the underlying manifold, we get potentially different bounded volume classes. The goal of this talk will be to explain how these bounded classes change as the (quasi-) isometry type of the hyperbolic structure changes. Along the way, we will contemplate the classification of Kleinian groups by their end invariants and explore some interesting properties of bounded cohomology.

Ilya Kapovich (CUNY Hunter College)

Title: Index properties of random automorphisms of free groups

Abstract: For automorphisms of the free group \(F_r\), being "fully irreducible" is the main analog of the property of being a pseudo-Anosov element of the mapping class group. It has been known, because of general results about random walks on groups acting on Gromov-hyperbolic spaces, that a "random" (in the sense of being generated by a long random walk) element \(\phi\) of \(\mathrm{Out}(F_r)\) is fully irreducible and atoroidal. But finer structural properties of such random fully irreducibles \(\phi\in \mathrm{Out}(F_r)\) have not been understood. We prove that for a "random" \(\phi\in \mathrm{Out}(F_r)\) (where \(r\ge 3\)), the attracting and repelling \(\mathbb R\)-trees of \(\phi\) are trivalent, that is all of their branch points have valency three, and that these trees are non-geometric (and thus have index \(<2r-2\)). The talk is based on a joint paper with Joseph Maher, Samuel Taylor and Catherine Pfaff.

Justin Lanier (Georgia Tech)

Title: Normal generators for mapping class groups are abundant

Abstract: For mapping class groups of surfaces, we provide a number of simple criteria that ensure that a mapping class is a normal generator, with normal closure equal to the whole group. We then apply these criteria to show that every nontrivial periodic mapping class that is not a hyperelliptic involution is a normal generator whenever genus is at least 3. We also show that every pseudo-Anosov mapping class with stretch factor less than √2 is a normal generator. Showing that pseudo-Anosov normal generators exist at all answers a question of Darren Long from 1986. In addition to discussing these results on normal generators, we will describe several ways in which they can be leveraged to answer other questions about mapping class groups. This is joint work with Dan Margalit.

Alexander Olshanskiy (Vanderbilt)

Title: Conjugacy problem in groups with quadratic Dehn function

Abstract: The minimal non-decreasing function \(d(n)\) such that every word \(w\) vanishing in a finitely presented group \(G= < A | R >\) and having length \(|w| ≤ n\) is freely equal to a product of at most \(d(n)\) conjugates of relators from \(R^{\pm 1}\), is called the Dehn function of the presentation \(< A | R >\) . In other words, the Dehn function \(d(n)\) of the presentation is the smallest function that bounds from above the areas of loops of length at most \(n\) in the Cayley complex \(Cay(G)\). Up to equivalence it does not depend on a finite presentation of \(G\). We construct a group with quadratic Dehn function and undecidable conjugacy problem. This solves E.Rips' problem formulated in 1992. (Joint work with Mark Sapir.)

Matthew Haulmark (Vanderbilt)

Title: TBA

Abstract: TBA