# Quasi-isometric groups

From Groupprops

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

## Definition

Two finitely generated groups are termed **quasi-isometric** if the following equivalent conditions hold:

- There exists a finite generating set for and a finite generating set for such that the Cayley graph for with respect to is quasi-isometric to the Cayley graph for with respect to .
- For every finite generating set for and every finite generating set for , the Cayley graph for with respect to is quasi-isometric to the Cayley graph for with respect to .

## Facts

- The relation of being quasi-isometric is an equivalence relation.
- Every finite group is quasi-isometric to the trivial group, and hence any two finite groups are quasi-isometric. In fact, a group is quasi-isometric to the trivial group if and only if it is finite.
- Every group is quasi-isometric to any subgroup of finite index in it.