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Ph.D. Program in Mathematics Upcoming Defense of Dissertations:

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Title: Hierarchical hyperbolicity of graph products and graph braid groups
Date, time, and location: April 6th, 2021, at 1pm.
Contact: Daniel Berlyne for zoom information.
Committee Members: 
Chair
:  Jason Behrstock (Lehman College & The Graduate Center)
Ilya Kapovich (Hunter College & The Graduate Center )
Joseph Maher (College of Staten Island & The Graduate Center )
Abstract: We show that any graph product of finitely generated groups is hierarchically hyperbolic relative to its vertex groups. We apply this result to answer two questions of Genevois about the geometry of 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 any graph product forms a hierarchically hyperbolic space, and that graph products of hierarchically hyperbolic groups are themselves hierarchically hyperbolic groups. To achieve this last result, we develop a technique which allows an almost hierarchically hyperbolic structure to be promoted to a hierarchically hyperbolic structure. 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, acylindrically hyperbolic, relatively hyperbolic, or thick.