Spring 2023

Abstract:  Gromov introduced some “hyperbolization” procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature, and Ontaneda provided a Riemannian version that works in the smooth setting. These procedures have been used to construct examples of manifolds and groups that exhibit various pathologies, despite having negative curvature. In joint work with J. Lafont, we construct actions of the fundamental groups of these spaces on CAT(0) cube complexes. As an application, we obtain that these groups are virtually special, hence linear over the integers and residually finite. In particular, we obtain new examples of negatively curved Riemannian manifolds whose fundamental groups virtually algebraically fiber.

Abstract: Consider a polygon-shaped billiard table in the hyperbolic plane on which a ball can roll along geodesics and reflect off of edges infinitely. In joint work with Viveka Erlandsson and Chris Leininger, we have characterized the relationship between the shape of the polygon and the set of possible sequences of edges visited consecutively by billiard balls rolling and reflecting around the polygon. In order to do this, we made an arguably more interesting characterization: when a hyperbolic metric with cone points on a surface is determined by the geodesics that do not pass through cone points. In this talk, we will explore these characterizations and the tools used to prove them.

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