Amalgamations and automorphism groups

Abstract

Many types of automorphism groups in algebra have nice structures arising from actions on combinatoric spaces. We recount some examples including Nagao's Theorem, the Jung-Van der Kulk Theorem, and a new structure theorem for the tame subgroup $\text{TA}_3(K)$ of the group $\text{GA}_3(K)$ of polynomial automorphisms of $\mathbb{A}_K^3$, for $K$ a field of characteristic zero. We also ask whether a larger collection of automorphism groups possess a similar kind of structure.