A note on quasi-transitive graphs quasi-isometric to planar (Cayley) graphs

TL;DR

This paper shows that any connected, locally finite, quasi-transitive graph quasi-isometric to a planar graph can be upgraded to a planar Cayley graph, addressing a question in geometric group theory.

Contribution

It proves that such graphs can be enhanced to planar Cayley graphs, bridging a gap in understanding quasi-isometric relations in planar graph theory.

Findings

Any such graph can be upgraded to a planar Cayley graph.

Addresses a previously open question in the field.

Links quasi-isometry classes to Cayley graph structures.

Abstract

Given a connected, locally finite, quasi-transitive graph $X$ which is quasi-isometric to a planar graph $\Gamma$, we remark that one can upgrade $\Gamma$ to be a planar Cayley graph, answering a question raised by Esperet--Giocanti and Hamann.