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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm111-1-13,
author = {N. Brodskiy and J. Dydak and A. Mitra},
title = {Coarse structures and group actions},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {149-158},
zbl = {1141.22007},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm111-1-13}
}
N. Brodskiy; J. Dydak; A. Mitra. Coarse structures and group actions. Colloquium Mathematicae, Tome 111 (2008) pp. 149-158. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm111-1-13/