Permanence in coarse geometry (Q2874865)

The development of coarse geometry has primarily been motivated by Gromov's work on geometric group theory and Yu's work on the Novikov Conjecture. With a number of applications at hand, the coarse geometry itself became an object of interest. This paper is a survey on permanence results in coarse geometry. By permanence we mean the preservation of coarse properties by various constructions.NEWLINENEWLINEThe paper begins with an introduction to coarse geometry suitable for non-experts as well as experts. It then presents the coarse properties whose permanence results are to be considered: finite asymptotic dimension, property \(A\), exactness and coarse amenability. The following permanence results are considered within the context of permanence for coarse metric spaces: coarse invariance, subspace permanence, union permanence, fibering permanence, and limit permanence. Since the main source of applications of coarse geometry is related to the induced coarse structure on finitely generated groups, a number of derived permanence results is also considered in the group setting. These include the permanence for direct unions, group extensions, free products, and others. Some results also discuss permanence in the context of group actions.NEWLINENEWLINEOverall, the paper is a systematic survey on results, which are usually scattered throughout the literature. When possible, the permanence results are proved via a broader axiomatic approach, resembling the categorical point of view, thus putting the presented permanence results into a broader context than the isolated permanence results previously found in the literature. Furthermore, such approach provides a framework which can be used for a systematical study of permanence results for other coarse properties.NEWLINENEWLINEFor the entire collection see [Zbl 1282.54001].