Commensurators and classifying spaces with virtually cyclic stabilizers (Q374097)

Let \(G\) be a discrete group and consider \({\underline{\underline{E}}}\) \(G\) the classifying space for the family of virtually cyclic subgroups, i.e. a CW-complex with an action of \(G\) such that the fixed points \(X^H\) are either empty -- if \(H\) is a subgroup of \(G\) which is not virtually cyclic -- or contractible -- if \(H\) is virtually cyclic. The main result of this article is a positive answer to the question of \textit{M. G. Fluch} and \textit{B. E. A. Nucinkis} [Proc. Am. Math. Soc. 141, No. 11, 3755--3769 (2013; Zbl 1341.55006)] whether every amenable group \(\Gamma\) of finite Hirsch length has a finite dimensional model for \(\underline{\underline{E}}\) \(\Gamma\).NEWLINENEWLINEThe strategy of the proof is to deal first with extensions of a locally finite group by a virtually cyclic one. To do so, a so-called strong equivalence relation is defined on infinite virtually cyclic subgroups, and a push-out construction of \textit{W. Lück} and \textit{M. Weiermann} [Pure Appl. Math. Q. 8, No. 2, 497--555 (2012; Zbl 1258.55011)] is applied to get information about the classifying space for virtually cyclic groups from the one for finite subgroups. This relies on the analysis of commensurators in \(\Gamma\) of infinite virtually subgroups (those elements \(x\) in \(\Gamma\) such that the intersection of the infinite virtually cyclic subgroup with its conjugate by \(x\) is infinite). The main theorem follows then from the structure of amenable groups of finite Hirsch length since any such group can be described by a couple of extensions where the other parts are free abelian, locally finite, and torsion-free nilpotent.