# Quasi-isometric groups

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 $G_1,G_2$ are termed quasi-isometric if the following equivalent conditions hold:

1. There exists a finite generating set $A_1$ for $G_1$ and a finite generating set $A_2$ for $G_2$ such that the Cayley graph for $G_1$ with respect to $A_1$ is quasi-isometric to the Cayley graph for $G_2$ with respect to $A_2$.
2. For every finite generating set $A_1$ for $G_1$ and every finite generating set $A_2$ for $G_2$, the Cayley graph for $G_1$ with respect to $A_1$ is quasi-isometric to the Cayley graph for $G_2$ with respect to $A_2$.

## 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.