Date of Degree

6-2020

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Jason Behrstock

Committee Members

Joseph Maher

Ilya Kapovich

Lee Mosher

Subject Categories

Geometry and Topology

Keywords

Hierarchically hyperbolic spaces, relative hyperbolicity, quasiconvexity

Abstract

Introduced by Behrstock, Hagen, and Sisto, hierarchically hyperbolic spaces axiomatized Masur and Minsky's powerful hierarchy machinery for the mapping class groups. The class of hierarchically hyperbolic spaces encompasses a number of important and seemingly distinct examples in geometric group theory including the mapping class group and Teichmueller space of a surface, virtually compact special groups, and the fundamental groups of 3-manifolds without Nil or Sol components. This generalization allows the geometry of all of these important examples to be studied simultaneously as well as providing a bridge for techniques from one area to be applied to another.

This thesis presents an introduction to the topic of hierarchically hyperbolic spaces as well as several original contributions by the author. Chapters 1 and 2 provide a brief introduction to the study of the coarse geometry of groups and spaces initiated by Gromov. Chapter 3 provides an introduction to hierarchically hyperbolic spaces and a summary of foundational results in the theory. Chapter 4 presents the original research of the author. These results largely focus on upstanding intertwined notions of convexity and curvature in hierarchically hyperbolic spaces.

The work in Chapter 4 contains four main results. The first result is a construction of hierarchically quasiconvex hulls for any subsets of a hierarchically hyperbolic space by iteratively connecting pairs of point by special quasi-geodesics called hierarchy paths. This construction mimics the construction of quasiconvex hulls in hyperbolic spaces by connecting pairs of points with geodesics and is an integral tool for our second result. The second result characterizes strongly quasiconvex subsets of hierarchically hyperbolic spaces in terms of their contracting behavior and the hierarchy structure. As an application of this result we prove that the hyperbolically embedded subgroups of a hierarchically hyperbolic group are precisely the almost malnormal, strongly quasiconvex subgroups. The third result proves that a simple, combinatorial condition called isolated orthogonality is sufficient for a hierarchically hyperbolic space to relatively hyperbolic. We apply this result to show that the separating curve graph of a closed surface or a surface with two punctures is relatively hyperbolic as well as recover results of Brock and Masur on the relative hyperbolicity of the Weil--Peterson metric on Teichmueller space for medium complexity surfaces. Our final result is a highly technical proof that the almost hierarchically hyperbolic spaces, introduced by Abbott, Behrstock, and Durham are all actually hierarchically hyperbolic spaces. This plugs a whole in the theory of hierarchically hyperbolic space discovered by Abbott, Behrstock, and Durham.

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