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: Non-hyperbolic groups with Menger curve boundary

Abstract: In the hyperbolic setting, groups with Menger curve boundary are known to be abundant. It was a surprising observation of Ruane that there were no known examples of non-hyperbolic groups with Menger curve boundary found in the literature. Thus Ruane posed the problem (early 2000's) of finding examples (alt. interesting classes) of non-hyperbolic groups with Menger curve boundary. In this talk I will discuss the first class of such examples. This is joint work with Chris Hruska and Bakul Sathaye. Time permitting, I will also discuss related work concerning right-angled Coxeter groups.

Sahana Balasubramanya (University of North Carolina, Greensboro)

Title: Hyperbolic structures on wreath products

Abstract: The poset of hyperbolic structures on a group G is still very far from being understood and several questions remain unanswered. In this talk, I will speak about some new results that describe hyperbolic structures on the wreath product Gwr Z, for any group G. As a consequence, I answer two open questions regarding quasi-parabolic structures: I will give an example of a group G with an uncountable chain of quasi-parabolic structures and give examples of groups that have finitely many quasi-parabolic structures.

Abdalrazzaq Zalloum (SUNY Buffalo)

Title: The growth of contracting geodesics in a finitely generated group

Abstract: The study of Gromov hyperbolic groups has been so fruitful that extending tools from this setting to more general classes of groups is a central theme in geometric group theory. The talk will be about our result with Joshua Eike stating that for any finitely generated group G, and any finite generating set A, the language consisting of all geodesics in Cay(G,A) with a "hyperbolicity condition" is a regular language. If G is hyperbolic itself, then all geodesics in Cay(G,A) will satisfy the hyperbolicity condition, therefore, our result recovers a classical result by James Cannon stating that the language of all geodesics in a hyperbolic group is a regular language. This in particular implies that for any finitely generated group, the growth function for the hyperbolic-like geodesics is rational. Time permitting, we will discuss an exciting potential application to the problem of counting pseudo-Anosovs in a mapping class group, and more generally, Morse elements in hierarchically hyperbolic groups.

Kevin Kordek (Georgia Tech)

Title: The rational homology of the level 4 braid group

Abstract: In this talk I will describe recent joint work with Dan Margalit on the rational homology of the level 4 braid group, a finite-index subgroup of the braid group, which is the kernel of the mod 4 reduction of the integral Burau representation. The main results are an explicit description of the first rational homology as a representation of the braid group and a formula for the first Betti number.

No Seminar This week. However, **Maria Gerasimova's** talk in the Subfactor Seminar on **February 8** may be of interest and will serve as an introduction to her talk in our seminar on February 20.

Bin Sun (Vanderbilt)

Title: Cohomology of group theoretic Dehn fillings

Abstract: The notion of a group theoretic Dehn filling generalizes that of a geometric Dehn filling of 3-manifolds. By refining the Lyndon-Hochschild-Serre spectral sequence, we obtain a spectral sequence to compute the cohomology of the quotient arising from a Dehn filling of a hyperbolically embedded subgroup. As an application, we estimate the cohomology dimension of the corresponding Dehn filling quotients. Moreover, we construct, for any two given finitely generated acylindrically hyperbolic groups, a finitely generated, acylindrically hyperbolic quotient with a nice bound on its cohomological dimension.

Maria Gerasimova (Technische Universität Dresden)

(Gerasimova will also speak on *Unitarisability of discrete groups* in the Subfactor Seminar on **February 8**)

Title: Asymptotics of Cheeger constants and unitarisability of groups

Abstract: Let \(\Gamma\) be a discrete group. A group \(\Gamma\) is called *unitarisable* if for any Hilbert space \(H\) and any uniformly bounded representation \(\pi: \Gamma \to B(H)\) of \(\Gamma\)
on \(H\) there exists an operator \(S: H\to H\) such that \(S^{-1}\pi(g)S\) is a unitary representation for every \(g \in \Gamma\). It is well known that amenable groups are unitarisable.
It has been open ever since whether amenability characterises unitarisability of groups.

One of the approaches to study unitarisability is related to the space of Littlewood functions \(T_1(\Gamma)\). We define the **Littlewood exponent** \({\rm Lit}(\Gamma)\) of a group \(\Gamma\):

We will also discuss some corollaries of this connection, for example, about the number of colours one\ need to colour the Cayley graphs of some groups.

Arie Levit (Yale)

Title: Surface groups are flexibly stable

Abstract: A group G is stable in permutations if every almost-action of G on a finite set is close to some actual action. Part of the interest in this notion comes from the observation that a non-residually finite stable group cannot be sofic.

I will show that surface groups are stable in a flexible sense, that is if one is allowed to "add a few extra points" to the permutation. This is the first non-trivial stability result for a non-amenable group.

The proof is essentially geometric. Along the way, we establish a quantitative variant of the LERF property for surface groups which may be of independent interest.

The talk is based on a joint work with Nir Lazarovich and Yair Minsky.

Yuri Bahturin (Memorial University of Newfoundland)

Title: Graded representations of simple Lie algebras

Abstract: We try to determine when and if an irreducible module V over a simple Lie algebra L over an algebraically closed field F of characteristic zero can be given a grading by an abelian group G, which is compatible with the grading of L by G. At this time, the classification of abelian group gradings on classical simple Lie algebras is complete, so switching to the representations looks like a logical next step.