Abstract
This chapter considers the Thompson's group F. Thompson's group F exhibits several behaviors that appear paradoxical. For example: F is finitely presented and contains a copy of F x F, indicating that F contains the direct sum of infinitely many copies of F. In addition, F has exponential growth but contains no free groups of rank 2. After providing an overview of the analytic definition and basic properties of the Thompson's group, the chapter introduces a combinatorial definition of F and two group presentations for F, an infinite one and a finite one. It also explores the subgroups, quotients, endomorphisms, and group action of F before concluding with an analysis of several geometric properties of F such as word length, distortion, dead ends, and growth. The discussion includes exercises and research projects.
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