Tree amalgamations and quasi-isometries

TL;DR

This paper explores how the quasi-isometry type of complex graphs can be understood through their simpler one-ended factors, extending known theorems from group theory to graph theory.

Contribution

It establishes that the quasi-isometry type of multi-ended accessible graphs is determined by their one-ended factors, generalizing existing results to a broader class of graphs.

Findings

Quasi-isometry type of multi-ended graphs determined by one-ended factors

Extension of Papsoglu and Whyte's theorems to graphs

Discussion on implications for Woess's question

Abstract

We investigate the connections between tree amalgamations and quasi-isometries. In particular, we prove that the quasi-isometry type of multi-ended accessible quasi-transitive connected locally finite graphs is determined by the quasi-isometry type of their one-ended factors in any of their terminal factorisations. Our results carry over theorems of Papsoglu and Whyte on quasi-isometries between multi-ended groups to those between multi-ended graphs. In the end, we discuss the impact of our results to a question of Woess.