On asymptotic dimension of groups acting on trees

TL;DR

This paper establishes an upper bound on the asymptotic dimension of groups acting on trees, specifically for fundamental groups of graphs of groups with finitely generated vertex groups, improving understanding of their large-scale geometric properties.

Contribution

It proves that the asymptotic dimension of such a fundamental group is at most one more than the maximum asymptotic dimension of its vertex groups, providing optimal estimates for HNN extensions and amalgamated products.

Findings

Asdim of the fundamental group is at most n+1.

Optimal estimate for HNN extensions.

Optimal estimate for amalgamated products.

Abstract

We prove the following theorem: Let $\pi$ be the fundamental group of a finite graph of groups with finitely generated vertex groups $G_v$ having asdim $G_v\le n$ for all vertices $v$. Then asdim$\pi\le n+1$. This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.