Twisted Lefschetz numbers of infra-solvmanifolds and algebraic groups (Q2285297)

Let \(M\) be an infra-solvmanifold with the fundamental group \(\Gamma\) and \(\varphi:M\to M\) be a diffeomorphism. Denote by \(f:\Gamma\to \Gamma\) the homomorphism induced by \(\phi\). For \(V_\phi\in\mathrm{Rep}(\Gamma)\), we define the flat bundle \(E_\phi=(\widetilde M\times V_\phi)/\Gamma\). Take \(V_\phi\in\mathrm{Rep}(\mathcal{A}_\Gamma/\mathcal{U}_\Gamma)\) and an isomorphism \(\Psi:V_{\phi\circ F}\to V_\phi\). Considering \(\Psi:V_{\phi\circ f}\to V_\phi\) as an isomorphism in \(\mathrm{Rep}(\Gamma)\), \(\Psi\) corresponds to an isomorphism \(\Xi:\varphi^*E_\phi\to E_\phi\) of flat bundles. The following is the main result in this paper: Theorem. Let \(A\) be a matrix representation of \(F_{\mathcal{U}_{\Gamma^*}}:u\to u\). Then, necessarily, \(L(\varphi,E_\phi,\Xi)=\mathrm{det}(I-A)\). Further aspects occasioned by these developments are also discussed.