On the coarse-geometric detection of subgroups

TL;DR

This paper explores how coarse geometric properties of Cayley graphs can identify subgroups, providing conditions for subgroup detection, invariance under quasi-isometries, and a partial converse to the Flat Torus Theorem for CAT(0) groups.

Contribution

It introduces coarse geometric criteria to detect subgroups in Cayley graphs and applies these to establish subgroup invariance and a partial Flat Torus Theorem converse.

Findings

Subgroups can be characterized by coarse geometric conditions.

Subgroups and splittings are invariant under quasi-isometries under certain conditions.

A partial converse to the Flat Torus Theorem is proved for CAT(0) groups.

Abstract

We generalize [Vav] to give sufficient conditions, primarily on coarse geometry, to ensure that a subset of a Cayley graph is a finite Hausdorff distance from a subgroup. Using this result, we prove a partial converse to the Flat Torus Theorem for CAT(0) groups. Also using this result, we give sufficient conditions for subgroups and splittings to be invariant under quasi-isometries.