Geometric Group Theory and Hyperbolic Groups

Geometric group theory is a relatively new field of mathematics that seeks to understand the connections between the algebraic properties of groups and the geometric or topological properties of the spaces on which they act. It is a field with opportunities for algebraic, geometric, topological, and combinatorial analysis. This thesis provides a concise and accessible introduction to the field of geometric group theory and hyperbolic groups. In this talk, I will first introduce some of the necessary background for a discussion of hyperbolic groups including topics on group presentations, Cayley graphs, and the insize definition of delta-hyperbolicity. I will then introduce a combinatorial problem (the word problem) and its application in geometric group theory. Finally, this talk will examine the word problem in hyperbolic groups and prove that hyperbolic groups always have solvable word problems because they admit Dehn presentations.