A counterexample to questions on the integrality property of virtual signature (Q1962099)

Let \(\Gamma\) be a group of finite homological type. The virtual Euler characteristic of \(\Gamma\) is defined as \(\chi(\Gamma)=\chi(\Gamma')/[\Gamma:\Gamma']\), where \(\Gamma'\subseteq\Gamma\) is any torsion free subgroup of finite index in \(\Gamma\). Under suitable finiteness conditions, K. Brown proved the following two results: Theorem 1. [\textit{K. S. Brown}, Invent. Math. 27, 229-264 (1974; Zbl 0294.20047)]. Let \(m\) be the least common multiple of the orders of finite subgroups of \(\Gamma\). Then \(m\cdot\chi(\Gamma)\) is an integer; and Theorem 2. [\textit{K. S. Brown}, op. cit.]. Let \(S\) denote the poset of nontrivial finite subgroups of \(\Gamma\) with \(\Gamma\) acting on \(S\) by conjugation, and \(|S|\) the associated simplicial complex, then \(\chi(\Gamma)\) is congruent to \(\chi_\Gamma(|S|)\bmod\mathbb{Z}\), where the latter is the equivariant Euler characteristic of \(|S|\). Let now \(\Gamma\) be a virtual Poincaré duality group (VPD), then the virtual signature of \(\Gamma\) (\(\text{Sgn}(\Gamma)\)) is defined and it is natural to ask whether results such as Theorems 1 and 2 are valid in this context. The author provides an example showing that this is false in general. More concretely, the author gives an example of a VPD group \(\Gamma\) such that (a) \(m\cdot\text{Sgn}(\Gamma)\) is not an integer and (b) \(\text{Sgn}(\Gamma)\not=0\), and \(\text{Sgn}(\Gamma)\) is not congruent to \(\text{Sgn}_\Gamma(|S|)\bmod\mathbb{Z}\).