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

Marek Golasiński,Daciberg Lima Gonçalves

doi:10.1007/s40062-016-0158-7

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.

Highlights

Recall that a finite dimensional C W -complex (n) with the homotopy type of the n-sphere Sn is called an n-homotopy sphere for n ≥ 1
We have described all virtually cyclic groups G that act on (2n) together with the induced action G → Aut(H 2n( (2n), Z)), and we classified the orbit spaces (2n)/G
An infinite virtually cyclic group is the middle term of a short exact sequence of the form e → Z → G → G0 → e, where G0 is a finite group

Summary

Introduction

Recall that a finite dimensional C W -complex (n) with the homotopy type of the n-sphere Sn is called an n-homotopy sphere for n ≥ 1. Observe that the results from [2] and [4] provide neither any information about the dimension n of the homotopy sphere of which the group G acts nor the induced action G → Aut(H n( (n), Z)), nor the homotopy type of orbit spaces. Studies these problems are relevant points of the present work. (2n − 1) and γ : (G1 ∗G0 G2) × (2n − 1) → (2n − 1) by the homotopy relation, are estimated in Corollary 4 and Corollary 6, respectively

Prerequisites

Actions of the group G Zm

Virtually cyclic groups

II III IV V VI