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The Ihara-Selberg zeta function of a tree lattice. (English) Zbl 0767.11025

This paper deals with a study of Ihara-Selberg zeta function of a finite graph. But his approach is slightly different from the one that has been usually done. He first defines the noncommutative determinant “\(\det_{P/A}(\alpha)\)”. Here \(A\) is a \(k\)-algebra, \(k\) a commutative ring containing the rational field \(\mathbb Q\), and \(\alpha\) is, say, a power series with constant term \(I\) with coefficients in \(\text{End}_ A(P)\). After developing general properties of determinants, the author proceeds to study the general theory of zeta functions of a uniform lattice on a tree. The zeta function is given as an “Euler product” \[ Z(u)=\prod_{\varepsilon\in\mathcal P} \det_{\mathbb C[\Gamma]/\mathbb C[\Gamma]}(1-\sigma_ \varepsilon u^{l_ \varepsilon})^{-1}. \] For a finite-dimensional representation \(\rho: \Gamma\to \text{GL}(V_ \rho)\), the \(L\)-function is defined by \(L(u,\rho)=\prod_{\varepsilon\in{\mathcal P}}\text{det}(I-\rho (\sigma_ \varepsilon) u^{l_ \varepsilon})^{-1}\). The author gives some properties of \(L(u,\rho)\) and the non-trivial zeros of \(L(u,\rho)\).
Reviewer: M.Muro (Yanagido)

MSC:

11M38 Zeta and \(L\)-functions in characteristic \(p\)
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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