Fall 2022

Abstract:  The homotopy type of the embedding space of a knot in R^3 or S^3 has been studied extensively, culminating in a complete description by Budney.  However, for links in R^3 much less is known. Brendle-Hatcher showed that the space of smooth embeddings of an n-component unlink in R^3 is homotopy equivalent to the space of round unlinks, i.e. where each component is a round circle lying in a plane.  In this talk we consider the embedding space of a split link with n sublinks, and show that the homotopy type of this embedding space depends only on that of the embeddings of a single sublink in a ball, together with a combinatorial object called the space of separating systems.  We use the space of separating systems to compute the fundamental group of the embedding space.  This is joint work with Rachael Boyd.

Abstract: The critical exponent is an important numerical invariant of discrete groups acting on negatively curved Hadamard manifolds, Gromov hyperbolic spaces, and higher-rank symmetric spaces. In this talk, I will focus on discrete groups acting on hyperbolic spaces (i.e., Kleinian groups), which is a family of important examples of these three types of spaces. In particular, I will review the classical result relating the critical exponent to the Hausdorff dimension using the Patterson-Sullivan theory and introduce new results about Kleinian groups with small or large critical exponents.

Abstract: The Gromov boundary provides a useful compactification for all infinite-diameter Gromov hyperbolic spaces. It consists of all geodesic rays starting at a given base-point and it has been an essential tool in the study of the coarse geometry of hyperbolic groups. In this study We find the natural analogue of the Gromov boundary for a larger class of metric spaces. We start in the setting of metric spaces that resemble the geometry of relatively hyperbolic groups. We define a notion of a boundary that is compact, metrizable and invariant under quasi-isometries. Furthermore, it contains the sublinearly Morse boundary as a dense, topological subspace. When the space in question has a cocompact group action, this boundary coincides with the Bowditch boundary. This is a joint work (in progress) with Kasra Rafi and Giulio Tiozzo.

Abstract: We consider the Weil-Petersson gradient vector field of renormalized volume on the deformation space of convex cocompact hyperbolic structures of a (relatively) acylindrical manifold. Using a toy model for the flow, we show that the flow has a global attracting fixed point at the structure M_geod the unique structure with totally geodesic convex core boundary. 
This is joint work with Kenneth Bromberg, and Franco Vargas Pallete.

Abstract: Given a finite simplicial graph, the associated right-angled Artin group (RAAG) is generated by all the vertices, and two generators commute if they are connected by an edge. The RAAG Recognition Problem asks whether a given group is (isomorphic to) a RAAG. In joint work with Lorenzo Ruffoni, we consider this recognition problem for Bestvina–Brady groups (BBGs). In this talk, I will describe a graphical condition to certify when a BBG is a RAAG. In particular, we will see a complete solution to the RAAG Recognition Problem for BBGs defined on 2-dimensional flag complexes.

Abstract: Self-similar groups arise as certain automorphism groups of trees. They are characterized by admitting a remarkable “recursive’’ decomposition for elements, and have been an important source of examples and counterexamples in geometric group theory. In this talk, we introduce shift-similar groups, which are groups of permutations of the natural numbers which are analogous in many ways to self-similar groups. We will discuss these analogies, as well as highlight how the two families differ: One particularly notable result is that there are uncountably many isomorphism classes of finitely generated shift-similar groups, in contrast with the self-similar case. This is joint work with Matthew Zaremsky (University at Albany).

Abstract: In this talk we will define what it means for a cube complex to be collapsible. In particular, our definition will apply to the case that the complex is not finite. Then, we will show that all locally-finite CAT(0) cube complexes are collapsible. The process will yield an inverse sequence of finite convex subcomplexes whose inverse limit provides a Z-compactification of the complex in which the boundary (which we call the cubical boundary) incorporates properties of both the visual and Roller boundaries.

Abstract: The central theme of geometric group theory is to study groups
via their actions on metric spaces. A model geometry of a finitely generated group is a proper geodesic metric space admitting a geometric group action. Every finitely generated group has a model geometry that is a locally finite graph, namely its Cayley graph with respect to a finite generating set. In this talk, I investigate which finitely generated groups G have the property that all model geometries of G are (essentially) locally finite graphs.

I introduce the notion of domination of metric spaces and give necessary and sufficient conditions for all model geometries of a finitely generated group to be dominated by a locally finite graph. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). Among groups of cohomological dimension two, the only such groups are surface groups and generalised Baumslag-Solitar groups.

Abstract: The Thurston Metric, introduced by Thurston in 1986, is an asymmetric metric on Teichmuller space, which measures distance between surfaces using the Lipschitz constant of maps between them. In this talk, I will tell you what I know about geodesics in this metric. Specifically, I will tell you about the geodesic envelope, its shape (and how the Thurston metric is similar to L^(infty)), and its width (and how the Thurston metric is not similar to L^(infty)). We’ll finish up with a theorem which gives sufficient conditions for geodesics between two points to be “essentially unique” (ie, uniformly bounded diameter from each other) for low complexity surfaces.

Abstract: In this talk, we will discuss subgroups of right-angled Artin groups (RAAGs for short). Although, in general, subgroups of RAAGs are known to have a wild structure and bad algorithmic behaviour, we will show that under certain conditions they have a tame structure. Firstly, we will discuss finitely generated normal subgroups of RAAGs and show that they are co-(virtually abelian). As a consequence, we deduce that they have decidable algorithmic problems. Secondly, we will recall results of Baumslag-Roseble and Bridson-Howie-Miller-Short on subgroups of direct products of free groups and explain how they generalise to other classes of RAAGs.