Coarse Equivalence and Topological Couplings of Locally Compact Groups

M. Gromov has shown that any two finitely generated groups Γ and Λ are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions Γ y X x Λ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second-countable groups. In his seminal monograph on geometric group theory [1], M. Gromov formulated a topological criterion for quasi-isometry of finitely generated groups (see 0.2.C′ 2 in [1]). His idea was to replace the geometric objects, that is, the finitely generated groups, by a purely topological framework, namely, a locally compact Hausdorff space, which has no intrinsic large scale geometric structure. As it is, Gromov’s proof easily adapts to characterise coarse equivalence of arbitrary countable discrete groups, but thus far the case of locally compact groups has not been adressed and indeed Gromov’s construction is insufficient to deal with these. The present paper presents a solution to this problem by establishing the following theorem. Theorem 1. Two locally compact second-countable groups are coarsely equivalent if and only if they admit a topological coupling. As coarse equivalence of locally compact, compactly generated groups is just quasi-isometry, we have the following corollary. Corollary 2. Two compactly generated, locally compact second-countable groups are quasi-isometric if and only if they admit a topological coupling. Let us recall that a topological coupling of two locally compact groups G and H is a pair G y X x H of commuting, proper and cocompact continuous actions on a non-empty locally compact Hausdorff space X. Here the actions are proper if, for every compact subset K ⊆ X, the sets {g ∈ G | gK ∩ K 6= ∅}, {h ∈ H | Kh ∩ K 6= ∅}