Colimits of abelian groups (Q497695)

Let \(G\) be a discrete group and let \(BG\) denote the classifying space (i.e. the geometric realization) of the simplicial set \(B_{\bullet}G\) given by \(B_nG=G^n\) with usual simplicial operations. For each positive integer \(q\), let \(B(q,G)\) denote the classifying space of the simplicial subset \(B_{\bullet}(q,G)\) of \(B_{\bullet}G\), where \(B_n(q,G)\) denotes the set of all \((g_1,\cdots ,g_n)\in G^n\) such that the subgroup \(\langle g_1,\cdots ,g_n\rangle\) generated by \(\{g_i\}_{i=1}^n\) is a subgroup of nilpotency class less than \(q\). The set \(\{g_i\}^{2r}_{i=1}\subset G\) of non-identity elements is called a symplectic sequence if \([g_i,g_{i+r}]=[g_j,g_{j*+r}]\) for all \(1\leq i,j\leq r\) and \([g_i,g_j]=1\) otherwise, where \([g,h]=ghg^{-1}h^{-1}.\) In this paper, the author studies the homotopy type of the space \(B(q,G)\) and he considers whether it is an Eilenberg-MacLane space of type of \(K(\pi,1)\). When \(G\) is a finite group and \(q=2\), he proves that \(B(2,q)\) is not an Eilenberg-MacLane space of type of \(K(\pi,1)\) if the group \(G\) has an non-trivial symplectic sequence \(\{g_i\}^{2r}_{i=1}\) for some \(r\geq 2\). As an application, he also obtains that extra-special \(p\)-groups of rank \(\geq 4\), general linear groups \(\text{GL}_n(\mathbb{F}_q)\) \((n\geq 4)\) and the symmetric group \(\Sigma_k\) of \(k\) letters \((k\geq 2^4)\) are not Eilenberg-MacLane spaces of type of \(K(\pi ,1)\).