Free and properly discontinuous actions of groups $$G\rtimes {\mathbb {Z}}^m$$ G ⋊ Z m and $$G_1*_{G_0}G_2$$ 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 Z m and G 1 * G 0 G 2 . In particular, homotopy types of orbits of (2n − 1)-spheres (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 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.

Poincaré duality complexes with highly connected universal cover

“…Since putting this paper on the arXiv in May 2016 we have learned that Theorem 23 and much of the subsequent discussion may also be found in the final section of [9]. The paper [9] also gives estimates of the number of homotopy types realizing a given fundamental group.…”

“…Since putting this paper on the arXiv in May 2016 we have learned that Theorem 23 and much of the subsequent discussion may also be found in the final section of [9]. The paper [9] also gives estimates of the number of homotopy types realizing a given fundamental group. However we have chosen to retain our independent treatment as it is brief and is a natural complement to our more substantial earlier results.…”