On Local Tameness of Certain Graphs of Groups

TL;DR

This paper proves that the fundamental group of certain graphs of groups with Noetherian edges and locally tame vertices is itself locally tame, extending to groups with specific JSJ-decompositions and flexible vertex groups.

Contribution

It establishes the local tameness of fundamental groups of graphs of groups under specified conditions, generalizing previous results in group theory.

Findings

Fundamental groups of graphs of groups with Noetherian edges are locally tame.

Groups with non-trivial JSJ-decomposition over VPC subgroups and flexible vertices are locally tame.

The result applies to a broad class of finitely presented groups.

Abstract

Let $G$ be the fundamental group of a finite graph of groups with Noetherian edges and locally tame vertices. We prove that $G$ is locally tame. It follows that if a finitely presented group $H$ has a non-trivial $JSJ$-decomposition over the class of its $VPC(k)$ subgroups for $k=1$ or $k=2$, and all the vertex groups in the decomposition are flexible, then $H$ is locally tame.