Asymptotic resemblance relations on groups

TL;DR

This paper explores how asymptotic resemblance relations induced by coarse structures on groups behave, generalizes asymptotic dimensiongrad, and introduces set theoretic coupling, establishing conditions for asymptotic equivalence.

Contribution

It introduces a generalized asymptotic dimensiongrad for groups with coarse structures and defines set theoretic coupling, extending topological coupling concepts.

Findings

Asymptotic dimensiongrad is coarse invariant.

Set theoretic coupling implies asymptotic equivalence.

Generalization of topological coupling for groups.

Abstract

In this paper, we study properties of asymptotic resemblance relations induced by compatible coarse structures on groups. We generalize the notion of asymptotic dimensiongrad for groups with compatible coarse structures and show this notion is coarse invariant. We end by defining the notion of set theoretic coupling for groups with compatible coarse structures and showing this notion is the generalization of the notion of topological coupling for finitely generated groups. We show if two groups with compatible coarse structures admit a set theoretic coupling then they are asymptotic equivalent.