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.