Asymptotic dimension of multi-ended quasi-transitive graphs

TL;DR

This paper establishes an upper bound on the asymptotic dimension of multi-ended quasi-transitive graphs formed via tree amalgamations, generalizing previous group-theoretic results to graph structures.

Contribution

It extends known bounds on asymptotic dimension from group theory to the setting of quasi-transitive graphs with tree amalgamations.

Findings

Upper bound on asymptotic dimension for tree amalgamations of graphs

Generalization of group-theoretic results to graph structures

Application to multi-ended quasi-transitive graphs

Abstract

We prove the existence of an upper bound on the asymptotic dimension of tree amalgamations of locally finite quasi-transitive connected graphs. This generalises a result of Dranishnikov for free products with amalgamation and a result of Tselekidis for HNN-extensions of groups to tree amalgamations of graphs. As a corollary, we obtain an upper bound on the asymptotic dimension of a multi-ended quasi-transitive locally finite graph based on any of their factorisations.