Geometry of non-transitive graphs

TL;DR

This paper investigates the geometric properties of non-transitive graphs under coarse transitivity conditions and constructs numerous non-quasi-isometric regular graphs sharing key geometric invariants with given groups.

Contribution

It introduces new results on the geometry of non-transitive graphs and constructs a large family of regular graphs with identical growth and asymptotic properties as specific groups.

Findings

Proves results on non-transitive graphs with coarse transitivity.

Constructs continuum many non-quasi-isometric regular graphs with the same invariants as a given group.

Abstract

In this note, we study non-transitive graphs and prove a number of results when they satisfy a coarse version of transitivity. Also, for each finitely generated group $G$, we produce continuum many pairwise non-quasi-isometric regular graphs that have the same growth rate, number of ends, and asymptotic dimension as $G$.