Quasi-isometric 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.