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Twisted Lefschetz numbers of infra-solvmanifolds and algebraic groups. (English) Zbl 1432.22014
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.
##### MSC:
 22E25 Nilpotent and solvable Lie groups 54H25 Fixed-point and coincidence theorems (topological aspects)
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##### References:
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