such as polynomial growth, the degree of polynomial growth, and exponential growth, are isomorphism invariants of groups. This topic is discussed further in the article on the growth rate of a group. Quasi-isometry invariants of a group In geometric group theory, groups are studied by their actions on metric spaces. A principle that generalizes the bilipschitz invariance of word metrics says that any finitely generated word metric on G is quasi-isometric to any proper, geodesic metric space on which G acts, properly discontinuously and cocompactly. Metric spaces on which G acts in this manner are called model spaces for G. It follows in turn that any quasi-isometrically invariant property satisfied by the word metric of G or by any model space of G is an isomorphism invariant of G. Modern geometric group theory is in large part the study of quasi-isometry invariants. See also Length function Longest element of a Coxeter group References J. W. Cannon, Geometric group theory, in Handbook of geometric topology pages 261--305, North-Holland, Amsterdam, 2002, Geometric group theory Metric geometry Properties of groups Combinatorics on words