Coarse structures and group actions

Abstract

The main results of the paper are: Proposition 0.1. A group G acting coarsely on a coarse space (X, C) induces a coarse equivalence g → g • x0 from G to X for any x0 Î A. Theorem 0.2. Two coarse structures C1 and C2 on the same set X are equivalent if the following conditions are satisfied: (1) Bounded sets in C1 are identical with bounded sets in C2. (2) There is a coarse action G_1 of a group G1 on (X,C1) and a coarse action ϕ2 of a group G2 on (A,C2) such that ϕ1 commutes with ϕ2. They generalize the following two basic results of coarse geometry: Proposition 0.3 (Shvarts Milnor lemma [5. Theorem 1.18]). A group G acting properly and cocompactly via isometrics on a length space X is finitely generated and induces a quasi-isometry equivalence g → g • xo from G to X for any x0 Î X. Theorem 0.4 (Gromov [4. p. 6]). Two finitely generated groups G and H are quasi-isometric if and only if there is a locally compact space X admitting proper and cocompact actions of both G and H that commute.