Dissertations, Theses, and Capstone Projects

Date of Degree

6-2021

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Jason Behrstock

Committee Members

Ilya Kapovich

Joseph Maher

Subject Categories

Geometry and Topology

Keywords

Geometric group theory, hierarchically hyperbolic group, cube complex, acylindrical hyperbolicity, relative hyperbolicity, thickness

Abstract

This thesis comprises three original contributions by the author concerning hierarchical hyperbolicity, a coarse geometric tool developed by Behrstock, Hagen, and Sisto to provide a common framework for studying aspects of non-positive curvature in a wide variety of groups and spaces.

We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this to answer two questions of Genevois about the electrification of a graph product of finite groups. We also answer two questions of Behrstock, Hagen, and Sisto: we show that the syllable metric on a graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. This last result is a strengthening of a result of Berlai and Robbio by removing the need for extra hypotheses on the vertex groups. To achieve this, we develop a technique that allows an almost hierarchically hyperbolic structure to be promoted to a hierarchically hyperbolic structure. This technique has found independent use in work of Abbott, Behrstock, and Durham, where it is used to significantly streamline their proofs.

We then turn to graph braid groups, using their structure as fundamental groups of special cube complexes to endow them with a natural hierarchically hyperbolic structure. By expressing this structure in terms of the graph, we obtain characterisations of when these groups are hyperbolic or acylindrically hyperbolic. We also conjecture and partially prove a similar characterisation of relative hyperbolicity.

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