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.

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