Quasi-isometric groups

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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.