On the virtually cyclic dimension of normally poly-free groups (Q6636828)

Let \(G\) be a group. A collection of subgroups of \(G\), say \(\mathcal{F}\), is called a \textit{family} if it is closed under taking subgroups and conjugation by elements of \(G\). Of special interest in this work are the families that consist of the trivial subgroup, of finite subgroups and of virtually cyclic subgroups of \(G\). Given a family \(\mathcal{F}\) there exists a universal space, denoted by \(E_{\mathcal{F}}G\), that classifies actions of \(G\) with isotropy in \(\mathcal{F}\). The \(\mathcal{F}\)-\textit{geometric} dimension of \(G\) is defined as the minimum dimension among such models for \(E_{\mathcal{F}}G\). We denote by \(gd(G),\ \underline{gd}(G)\) and \(\underline{\underline{gd}}(G)\) the geometric dimension for the trivial family, the family of finite subgroups and the family of virtually cyclic subgroups of \(G\), respectively. A group \(G\) is called \textit{poly-free} if there exists a finite filtration \(1=G_0\triangleleft G_1\triangleleft \cdots \triangleleft G_{n-1}\triangleleft G_n=G\) of subgroups of \(G\) such that \(G_{i+1}/G_i\) is a free group for \(i=1,\ldots ,n-1\). The group is called \textit{normally} poly-free if such a sequence exists with all \(G_i\triangleleft G\). The minimum length of such a sequence is called the \textit{length} of \(G\). The authors relate the various geometric dimensions of \(G\) with that of being poly-free or normally poly-free. The main result is the following \N\N\textbf{Theorem}. Let \(G\) be a poly-free group of length \(n\). Then, (a) \(gd(G)= \underline{gd}(G)\leq n\). Moreover, if \(G\) is normally poly-f.g.-free, then \(gd(G)=n\). (b) If \(G\) is normally poly-free, then \(\underline{\underline{gd}}\leq 3(n-1)+2\).