UB Geometry and Topology Seminar 

Abstract: Asymptotic dimension is a quasi-isometry invariant of a metric space, introduced by Gromov in 1999 as a large-scale analogue of the Lebesgue covering dimension (or topological dimension) of a topological space. Groups with finite asymptotic dimension are known to satisfy some nice properties, and all hyperbolic groups have finite asymptotic dimension. In this talk we will discuss how asymptotic dimension behaves under certain group-theoretic operations, some known relations between asymptotic dimension and other notions of dimension, and techniques for providing upper and lower bounds on the asymptotic dimension of a metric space. We also present the new result that all groups with presentations satisfying the C’(1/6) small cancellation condition have asymptotic dimension at most 2.

Abstract: The action of the mapping class group on Teichmüller space is central to our understanding of hyperbolic surfaces. Many of the same techniques have been used to study (outer) automorphism groups of free groups, using the action of the group on Culler-Vogtmann’s Outer Space. In this talk, I will present some current work with Bregman and Vogtmann on constructing a more general version of Outer Space to study automorphism groups of right-angled Artin groups.

Abstract: We investigate the manifold structure and the relatively hyperbolic structure of right-angled Coxeter groups with planar nerves. We use these structures to study the quasi-isometry problem for this class of right-angled Coxeter groups. We also give necessary and sufficient conditions for our groups to be quasi-isometric to a right-angled Artin group. This is a joint work with Matthew Haulmark and Hoang Thanh Nguyen.

Abstract: We show that a conjecture of Ageev follows from the Borsuk-Ulam-type conjecture of Baum, Dabrowski and Hajac. Then we explain how the Ageev conjecture implies the weak version of the Hilbert-Smith conjecture which states that no infinite compact zero-dimensional group can act freely on a manifold so that the orbit space is finite dimensional. The Hilbert-Smith conjecture originates from the already settled Hilbert's fifth problem concerning a characterization of Lie groups.

Abstract: Let $G$ be a finitely generated group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. A natural question is the following: what are the possible asymptotic behaviors for the length of $f^n(g)$, written as a word in the generating set $S$, as $n$ goes to infinity, and as $g$ varies in the group $G$? We investigate this question in the case where $G$ is a torsion-free Gromov hyperbolic group. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel’s work on train-tracks when $G$ is a free group. We address the case of a general torsion-free hyperbolic group. We show in particular that every element $g$ has a well-defined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$. This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.

Abstract: This is a joint project with Tom Goodwillie extending earlier joint work with Goodwillie and Ohrt. The purpose of this project is to construct all exotic smooth structures on all smooth manifold bundles with a fiberwise group action in a stable range of dimensions. I will start by defining and enumerating the exotic smooth structures. By ``enumerate'' I mean compute the dimension of the vector space of exotic structures. To do this, we use a simplified version of Mackey functors. We give a simple construction of these exotic structures using the irreducible real representations of a finite group $G$. We call it the ``equivariant Hatcher construction'' since it generalizes a classical construction due to Hatcher. We use higher Reidemeister torsion with coefficients in a Mackey functor to demonstrate that our construction (with all possible inputs) spans the vector space of all stable exotic structures on a fixed $G$-bundle.

Abstract: We establish a connection between extreme values of stable random fields arising in probability and groups $G$ acting geometrically on CAT(-1) spaces $X$. The connection is mediated by the action of the group on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value theory, we introduce an invariant of the action called extremal cocycle growth and show that its non-vanishing is equivalent to finiteness of the Bowen-Margulis measure for the associated unit tangent bundle $U(X/G)$ provided $X$ is not a tree whose edges are (up to scale) integers. We also establish an analogous statement for normal subgroups of free groups. This is joint work with Parthanil Roy.

Abstract: A surface $S$ is of finite-type if its fundamental group is finitely generated; otherwise, it is of infinite type. The mapping class group MCG($S$) of $S$ is the group of isotopy classes of orientation-preserving homeomorphisms of $S$. This is a well-studied group when $S$ has finite type, but big mapping class groups, i.e. MCG($S$) of infinite-type surfaces, remain quite mysterious. But big mapping class groups arise naturally in various areas of mathematics and recently there has been a surge of interests in studying them. In this talk, I will discuss some recent results about the Torelli subgroup of MCG($S$). This is joint with Aramayona, Ghaswala, Kent, McLeay, and Winarski.

Abstract: It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. In this talk, we will discuss some new results about positive contact surgeries and in particular completely characterize when contact ($r$)-surgery is symplectically/Stein fillable for $r\in(0,1]$. This is joint work with James Conway and John Etnyre.

Abstract: Convexity is a fundamental notion across a variety of flavors of geometry. In the study of the course geometry of metric spaces, it is natural to study quasiconvexity i.e. convexity with respect to quasi-geodesics. We study quasiconvexity in the class of hierarchically hyperbolic spaces; a generalization of Gromov hyperbolic spaces which contains the mapping class group, right-angled Artin and Coxeter groups, and many 3-manifold groups. ​ Inspired by the rich theory of quasiconvexity in hyperbolic spaces, we show that quasiconvex subsets of hierarchcially hyperbolic spaces mimic the behavior of quasiconvex subsets in hyperbolic spaces.