| Summary: | Geometric group theory refers to the study of finitely generated groups and their properties by exploring the algebraic and topological structure. This thesis will look at various enumeration problems that arises in Baumslag-Solitar groups. Initially, this thesis aims to reproduce and validate some of the work that has been done on the questions of growth, cogrowth and geodesic elements via an enumeration approach. This approach will then be used to explore specific examples of Baumslag-Solitar groups where these questions have not been fully answered.
The first part of this thesis will look at the growth of a horocyclic subgroup in Baumslag-Solitar groups. It will then build upon this to develop an algorithm to count the elements of the group in general out to a fixed radius with the intention of further understanding the unresolved cases.
The second part of this thesis will use Baumslag-Solitar groups as a basis to develop numerical tests to estimate cogrowth of groups. Since the cogrowth of a group is directly related to the amenability of the group, these numerical tests for cogrowth can be applied to groups such that Thompson's group F, where the question of amenability is still highly debated.
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