The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups

Abstract

We give a practical criterion to determine whether a given pair of morphisms between almost-crystallographic groups has a finite Reidemeister coincidence number. As an application, we determine all two- and three-dimensional almost-crystallographic groups that have the R∞ property. We also show that for a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, the Nielsen coincidence number equals the Reidemeister coincidence number when the latter is finite.