Use in Geometric Group Theory
Given a finite generating set S of a finitely generated group G, we can form the corresponding Cayley graph of S and G. This graph becomes a metric space if we declare the length of each edge to be 1. Taking a different finite generating set T results in a different graph and a different metric space, however the two spaces are quasi-isometric. This quasi-isometry class is thus an invariant of the group G. Any property of metric spaces that only depends on a space's quasi-isometry class immediately yields another invariant of groups, opening the field of group theory to geometric methods.
Read more about this topic: Quasi-isometry
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