Preserving Coarse Properties

TL;DR

This paper studies how coarse properties like asymptotic dimension are preserved under specific quotient maps induced by group actions, providing new characterizations and equivalences among these properties.

Contribution

It offers an alternative description of asymptotic Property C and proves the equivalence between straight finite decomposition complexity and countable asymptotic dimension.

Findings

Spaces with straight finite decomposition complexity are the same as those with countable asymptotic dimension.

Provides an alternative characterization of asymptotic Property C.

Shows preservation of coarse properties under quotient maps induced by group actions.

Abstract

The aim of this paper is to investigate properties preserved and co-preserved by coarsely $n$-to-1 functions, in particular by the quotient maps $X\to X/\sim$ induced by a finite group $G$ acting by isometries on a metric space $X$. The coarse properties we are mainly interested in are related to asymptotic dimension and its generalizations: having finite asymptotic dimension, asymptotic Property C, straight finite decomposition complexity, countable asymptotic dimension, and metric sparsification property. We provide an alternative description of asymptotic Property C and we prove that the class of spaces with straight finite decomposition complexity coincides with the class of spaces of countable asymptotic dimension.