Arithmetic manifolds are hyperbolic manifolds constructed from number theoretic data. By their very definition, they exhibit a strong connection between algebraic invariants, such as trace fields, and geometric quantities like lengths of closed geodesics. Despite this, the geometry of these manifolds remains surprisingly mysterious. Nevertheless, a guiding philosophy is that arithmetic manifolds should be the most symmetric hyperbolic manifolds and therefore exhibit geometric phenomena that are rare or absent in generic hyperbolic manifolds. In this talk, I will survey arithmetic hyperbolic manifolds, likely focusing on low dimensions, and discuss several manifestations of this philosophy, both known and conjectural. I will then discuss new work furthering this philosophy by establishing finiteness of closed arithmetic surface bundles, resolving a conjecture of Bowditch, Maclachlan, and Reid.
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In this talk, we will introduce the notion of an end-periodic homeomorphism of an infinite-type surface. We will explore how the geometry of the associated mapping torus is related to certain topological and dynamical features of the end-periodic gluing map. In particular, we will see how the hyperbolic volume of the 3-manifold can be bounded both above and below in terms of a certain dynamical feature of the homeomorphism. This talk represents joint work with Autumn Kent, Heejoung Kim, Christopher Leininger, and Marissa Loving (in various configurations)
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Right-angled Coxeter groups form an extremely accessible, yet remarkably rich class of objects in geometric group theory. They are defined by simple presentations: they are generated by involutions, with the only additional relations requiring certain pairs of generators to commute. Despite this elementary definition, they display an extraordinary range of geometric behaviors. Consequently, they have played a crucial role in the field, as a source of illuminating examples and counterexamples and as a testing ground for conjectures. In this talk, I will survey recent progress in understanding their large-scale geometry, focusing in particular on questions of quasi-isometry and commensurability. Along the way, I will illustrate some of the main tools and techniques used for establishing such results, many of which are applicable in more general settings.
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