Type of Document Dissertation Author Hassan Balasubramanya, Sahana Author's Email Address hbsahana@gmail.com, sahana.balasubramanya@vanderbilt.edu URN etd-05252018-123711 Title Hyperbolic Structures on groups Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Denis Osin Committee Chair Jesse Peterson Committee Member Mark Sapir Committee Member Mike Mihalik Committee Member Paul D.Sheldon Committee Member Keywords

- Quasi-parabolic actions
- Geometric group theory
- Group actions on hyperbolic spaces
- Acylindrical actions
- Lamplighter groups
Date of Defense 2018-04-25 Availability unrestricted AbstractIt is customary in geometric group theory to study groups as metric spaces. The standardway to convert a group G into a geometric object is to fix a generating set X and

endow the Cayley graph Ga(G,X) with the corresponding word metric. In joint work with Carolyn Abbott and Denis Osin, we introduced the set of hyperbolic structures on G, denoted H(G), which consists of equivalence classes of generating sets of G such that the corresponding Cayley graph is hyperbolic; these are ordered in a natural way according to the amount of information they provide about the group. Of special interest is the subset AH(G) of H(G) of acylindrically hyperbolic structures on G, i.e. hyperbolic structures corresponding to acylindrical actions.The question of accessibility of these posets is studied, and several classes of acylindrically hyperbolic groups are proved to be AH-accessible.

By utilizing the notions of hyperbolically embedded subgroups and projection complexes, I then prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph Ga(G,X) is a (non-elementary) quasi-tree and the action of G on Ga(G,X) is acylindrical. As an application, new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups are obtained.

Lastly, a particular question associated to quasi-parabolic hyperbolic structures is answered . Specifically, many examples of groups with finitely many quasi-parabolic structures are given.

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