Director of Research (if dissertation) or Advisor (if thesis)
Leininger, Christopher
Doctoral Committee Chair(s)
Dunfield, Nathan
Committee Member(s)
Hirani, Anil
Sadanand, Chandrika
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Geometric Group Theory
Handlebody Groups
Language
eng
Abstract
The main result of this thesis is that a finitely generated subgroup of the genus two handlebody group is stable if and only if the orbit map to the disk graph is a quasi-isometric embedding. To this end, we prove that the genus two handlebody group is a hierarchically hyperbolic group, and that the maximal hyperbolic space in the hierarchy is quasi-isometric to the disk graph of a genus two handlebody by appealing to a construction of Hamenstadt-Hensel. We then utilize the characterization of stable subgroups of hierarchically hyperbolic groups provided by Abbott-Behrstock-Durham. We also present several applications of the main theorems, and show that the higher genus analogues of the genus two results do not hold.
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