Gromov's Corollary, aka the Word Metric

We have seen a number of examples of groups acting on the real line. Sometimes these actions preserve the distance between points on the line, such as the action of D∞. Other actions we have considered do not preserve distances, for example, the action BS(1, 2) ↷ ℝ presented in Chapter 4. One of the most powerful insights in the study of finitely generated infinite groups is that they can always be viewed as groups acting in a distance-preserving way on a geometric object. We refer to this insight as Gromov's Corollary to Cayley's Better Theorem, in honor of a groundbreaking paper that Mikhail Gromov wrote in the 1980s [Gr87], which highlighted this perspective, introduced a number of questions motivated by the geometry of infinite groups, and introduced powerful tools that can be used to answer them.

In order to present Gromov's Corollary, we need to introduce a reasonably flexible notion of “geometric object” as well as formally define what we mean by saying that a group action “preserves distances.”