Coarse coherence of metric spaces and groups and its permanence properties

TL;DR

This paper introduces coarse coherence properties for metric spaces and groups, showing they are invariants under quasi-isometry and establishing permanence properties that aid in understanding their algebraic K-theory computations.

Contribution

It defines coarse coherence and coarse regular coherence as geometric counterparts to algebraic coherence, and proves their permanence properties, including fibering permanence.

Findings

Coarse regular coherence implies weak regular coherence.

All known weakly regular coherent groups are coarsely regular coherent.

Permanence properties are established for coarse regular coherence.

Abstract

We introduce properties of metric spaces and, specifically, finitely generated groups with word metrics which we call coarse coherence and coarse regular coherence. They are geometric counterparts of the classical algebraic notion of coherence and the regular coherence property of groups defined and studied by F. Waldhausen. The new properties can be defined in the general context of coarse metric geometry and are coarse invariants. In particular, they are quasi-isometry invariants of spaces and groups. We show that coarse regular coherence implies weak regular coherence, a weakening of regular coherence by G. Carlsson and the first author. The latter was introduced with the same goal as Waldhausen's, in order to perform computations of algebraic K-theory of group rings. However, all groups known to be weakly regular coherent are also coarsely regular coherent. The new framework allows us to prove structural results by developing permanence properties, including the particularly important fibering permanence property, for coarse regular coherence.