A twisted invariant for finitely presentable groups (Q5928557)

Assume that a finitely generated group \(\Gamma\) allows a homomorphism onto a cyclic group of order \(m\), then there is an associated homomorphism \(\alpha\) of the group ring \(\mathbb{Z}[\Gamma]\) onto the cyclotomic integers \(\mathbb{Z} [\zeta_m]\), \(\zeta_m\) a primitive root of unity. It is assumed that \(\mathbb{Z} [\zeta_m]\) is a unique factorization domain which is the case when the class number of the corresponding cyclotomic field is one. If in addition a representation \(\rho: \Gamma\to GL(n,\mathbb{Z})\) is given, Wada's construction of a twisted Alexander polynomial can be applied [\textit{M. Wada}, Topology 33, No. 2, 241-256 (1994; Zbl 0822.57006)]. The author proves via Tietze transformations that the construction does not depend on the presentation of \(\Gamma\) -- though, of course, it depends on \(\rho\). Some examples are given.