Free and properly discontinuous actions of groups \(G\rtimes {\mathbb {Z}}^m\) and \(G_1*_{G_0}G_2\)

Abstract

We estimate the number of homotopy types of orbit spaces for all free and properly discontinuous cellular actions of groups \(G\rtimes {\mathbb {Z}}^m\) and \(G_1*_{G_0}G_2\). In particular, homotopy types of orbits of \((2n-1)\)-spheres \(\Sigma (2n-1)\) for such actions are analysed, provided the groups \(G_0, G_1, G_2\) and G are finite and periodic. This family of groups \(G\rtimes {\mathbb {Z}}^m\) and \(G_1*_{G_0}G_2\) contains properly the family of virtually cyclic groups. The possible actions of those groups on the top cohomology of the homotopy sphere are determined as well.