Discrete Algebraic Methods

8 Discrete infinite groups

This chapter concerns some fundamental issues in combinatorial group theory with a focus on discrete infinite groups. The idea is to prepare the reader for more advanced books such as [7, 8, 15] or the classics like [69] or [93]. In particular, we discuss amalgamated products, HNN extensions, and their normal forms. We cover free groups and their rational subsets, and show that automorphism groups of finitely generated free groups are finitely generated themselves. The proof is based on simple transformations of finite edge-labeled graphs, which have an interpretation in terms of Whitehead automorphisms. Finally, we take a quick look at geometric aspects by determining the algebraic structure of SL(2, ℤ) as the amalgamated ...