Relative accessibility for graphs

TL;DR

This paper generalizes the concept of accessibility in graphs relative to peripheral systems, linking it to algebraic structures and invariance under quasi-isometries, with applications to finitely generated groups.

Contribution

It introduces a new relative notion of accessibility for graphs, characterizes it algebraically, and proves its invariance under certain quasi-isometries, extending existing theories.

Findings

Relative accessibility is a quasi-isometry invariant for finitely generated groups.

The new definition aligns with algebraic notions in group theory.

A relative version of Hamann's accessibility theorem is established.

Abstract

We relativise the Thomassen--Woess definition of accessibility in graphs, defining what it means for a graph to be accessible relative to a peripheral system. In the case of locally finite, quasi-transitive graphs, we characterise relative accessibility in terms of a certain subring of the Boolean ring of the graph, and apply this to show that our definition agrees with the usual algebraic notion of relative accessibility in finitely generated groups. This implies, in particular, that relative accessibility is a quasi-isometry invariant amongst finitely generated groups, when the quasi-isometry coarsely preserves the left cosets of the peripheral subgroups. We also deduce a relative variant of Hamann's accessibility theorem on graphs with finitely generated cycle spaces.