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 ∈ X.
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 ϕ₁ of a group G₁ on (X,𝓒₁) and a coarse action ϕ₂ of a group G₂ on (X,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 isometries on a length space X is finitely generated and induces a quasi-isometry equivalence g ↦ g·x₀ from G to X for any x₀ ∈ 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.