The asymptotic dimension of quotients by finite groups

Abstract

Abstract. Let X be a proper metric space and let F be a finite group acting on X by isometries. We show that the asymptotic dimension of F \X is the same as the asymptotic dimension of X.

Higson compactification and dimension raising

“…We also introduce coarsely open maps, which correspond to maps for which the induced map on the Higson corona is open, and prove that they preserve the asymptotic dimension in case they are coarsely n-to-1. They also contain the class of quotient maps Z → G\Z by the finite group action and as a corollary we deduce the main result of [20]. For another application of the Higson corona in coarse geometry see [16].…”

“…We use notation G \ X to denote the space of orbits {G · x} x∈X equipped with the Hausdorff metric. The notation is used in [20] and distinguishes the space of orbits from the quotient X/G (which denotes the 'right' orbits) in case when X is a group and G is its subgroup. However, in [14] the notation X/G is used (instead of G \ X) to denote the space of orbits as there is no reference to the subgroup case.…”

“…As in Example 4.2 of [14] we may assume that G acts on X by isometries. According to Proposition 3.1 of [20] G induces an action on νX and the map induced by X → G \ X is the quotient map νX → G \ νX, which is open. By…”

“…Our method of proof is to show that coarsely n-to-1 maps induce close n-to-1 maps of Higson Coronas and use a topological version of the Hurewicz dimension lowering theorem, alongside with the result of Dranishnikov, Keesling, and Uspenskij ( [10], [9]) that the covering dimension of the Higson corona of a proper metric space is equal to its asymptotic dimension. A similar approach was recently independently used by Kasprowski [20] in order to prove that if a finite group G is acting on a proper metric space Z by isometries then the quotient Z → G\Z preserves the asymptotic dimension: in this case the induced map on Higson corona is open and one may use the fact that open n-to-1 maps preserve dimension. We note that the class of coarsely n-to-1 maps is vastly larger that the class of quotient maps Z → G\Z by the finite group action (see [24] or Example 2.7 for an example of a coarsely n-to-1 maps strictly raising the asymptotic dimension).…”

Injectivity results for coarse homology theories

“…Let H denote the subgroup of G generated by all elements of distance at most R from the neutral element. Since H is a finitely generated abelian group of rank at most n, it has asymptotic dimension at most n. Moreover, there is an upper bound on the cardinality of the finite subgroups of H. Hence by [24,Corollary 1.2], the family {F \H} F ∈Fin(H) has asymptotic dimension at most n. In particular, there is S > 0 such that for every F \H there is a cover U F 0 ∪ . .…”