Coarse classification of abelian groups and amenable shift-homogeneous metric spaces

TL;DR

This paper classifies countable locally finite-by-abelian groups using a coarse classification of amenable shift-homogeneous metric spaces, providing a new framework for understanding their large-scale geometric structure.

Contribution

It introduces a coarse classification method for amenable shift-homogeneous metric spaces and applies it to classify certain abelian-related groups.

Findings

Countable locally finite-by-abelian groups are classified up to coarse isomorphism.

A coarse classification of amenable shift-homogeneous metric spaces is developed.

The classification links geometric properties of spaces to algebraic properties of groups.

Abstract

In this paper we classify countable locally finite-by-abelian groups up to coarse isomorphism. This classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.