Summary: “We study the fundamental group of the \(p\)-subgroup complex of a finite group \(G\). We show first that \(\pi_1 ( \mathcal{A}_3 ( A_{10} ) )\) is not a free group (here \(A_{10}\) is the alternating group on ten letters). This is the first concrete example in the literature of a \(p\)-subgroup complex with non-free fundamental group. We prove that, modulo a well-known conjecture of Aschbacher, \( \pi_1 ( \mathcal{A}_p ( G ) ) = \pi_1 ( \mathcal{A}_p ( S_G ) ) \ast F\), where \(F\) is a free group and \(\pi_1 ( \mathcal{A}_p ( S_G ) )\) is free if \(S_G\) is not almost simple. Here \(S_G = \Omega_1 ( G ) / O_{p^\prime} ( \Omega_1 ( G ) )\). This result essentially reduces the study of the fundamental group of \(p\)-subgroup complexes to the almost simple case. We also exhibit various families of almost simple groups whose \(p\)-subgroup complexes have free fundamental group.”
With GAP programs the authors compute \( \pi_1 ( \mathcal{A}_3 ( A_{10} )) \), the fundamental group of the geometric realization of the poset of \(3\)-subgroups of \( A_{10} \). This fundamental group is a free product of the free group on 25200 generators and a non-free group on 42 generators and 861 relators whose abelianization is \(\mathbb Z^{42}\). There are no torsion elements in \( \pi_1 ( \mathcal{A}_3 ( A_{10} )) \). The integral homology of \( \mathcal{A}_3 ( A_{10} )\) is known to free abelian. Their Theorem 5.1 allows to construct an infinite number of examples of finite groups \(G\) with non-free \( \pi_1 ( \mathcal{A}_3 ( G ) )\). But other theorems in the paper show that such examples are rather exceptional.