Master's lecture in Mathematics - Áslaug Haraldsdóttir

Master's student: Áslaug Haraldsdóttir

Title: Geometric properties of group actions on hyperbolic graphs

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Faculty: Faculty of Physical Sciences

Advisor:  Rögnvaldur G. Möller

Also in the masters committee: Jón Ingólfur Magnússon

Examiner: Henning Arnór Úlfarsson, Assistant Professor in Math at Reykjavik University

Abstract

This thesis is a study of group actions on connected, locally finite hyperbolic graphs. The main contribution of this work is to find a new way to show that a locally finite connected hyperbolic graph with a transitive group of automorphisms fixing a boundary point is quasi-isomorphic to a tree and thereby answer a question of Kaimanovich and Woess. This is done by investigating properties of hyperbolic metric spaces as well as using Cayley–Abels graphs and tidy subgroups. These results are then used to study certain subgraphs spanned by geodeisc lines. Finally it is shown that if a group acts both transitively on the vertex set and the boundary set of such a graph then the graph is quasi-isometric to a tree.