The type of the classifying space for a family of subgroups
Given a group \(\Gamma\), it is a classical problem to determine whether \(\Gamma\) can act freely and preserving cells on some contractible CW-space \(E \Gamma\) whose orbit space is finite, resp. finite dimensional, resp. of finite type. More generally, the author asks if there is a CW-space \({\mathcal E}\Gamma\) on which \(\Gamma\) acts with finite stabilizer groups and contractible fixed point sets and whose orbit space is finite, resp. finite dimensional, resp. of finite type. He gives a necessary condition for the existence of such \({\mathcal E}\Gamma\) such that the fixed point sets have a prescribed dimension. He also shows that this condition is sufficient whenever the length of descending sequences of finite subgroups in \(\Gamma\) is bounded. The existence question of \({\mathcal E}\Gamma\) such that \({\mathcal E}\Gamma/\Gamma\) is finite or of finite type is considered in several settings. The paper closes with a list of interesting open problems on this topic.
scientific article; zbMATH DE number 1474722