This chapter considers the notion of quasi-isometry, also known as “coarse isometry.” A whole suite of important algebraic and geometric properties is preserved by quasi-isometries. Quasi-isometry can be applied to the algebraic structure of groups. A sample result, which shows that quasi-isometries can have powerful algebraic consequences, is a theorem of Gromov. Along the way to this theorem, the chapter proves the Milnor–Schwarz lemma, sometimes referred to as the fundamental lemma of geometric group theory. After describing Cayley graphs as well as path metrics and word metrics for integers, the chapter explores the bi-Lipschitz equivalence of word metrics, quasi-isometric equivalence of Cayley graphs, quasi-isometries between groups and spaces, and quasi-isometric rigidity. The discussion includes exercises and research projects.
Lipschitz Equivalence of Self-Similar Sets: Algebraic and Geometric Properties