In 1911, mathematician Max Dehn posed three decision problems for finitely
presented groups that have remained central to the study of combinatorial
group theory. His work provided the foundation for geometric group theory,
which aims to analyze groups using the topological and geometric properties
of the spaces they act on. In this thesis, we study group actions on Cayley
graphs and the Farey tree. We prove that a group has a solvable word problem
if and only if its associated Cayley graph is constructible. Moreover, we prove
that a group is finitely generated if and only if it acts geometrically on a proper
path-connected metric space. As an example, we show that SL(2, Z) is finitely
generated by proving that it acts geometrically on the Farey tree.
| Identifer | oai:union.ndltd.org:CALPOLY/oai:digitalcommons.calpoly.edu:theses-4508 |
| Date | 01 June 2024 |
| Creators | LaBrie, Noelle |
| Publisher | DigitalCommons@CalPoly |
| Source Sets | California Polytechnic State University |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | Master's Theses |
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