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Zeta functions of finite Schreier graphs and their zig zag products (1508.00371v3)
Published 3 Aug 2015 in math.GR
Abstract: We investigate the Ihara zeta functions of finite Schreier graphs $\Gamma_n$ of the Basilica group. We show that $\Gamma_{1+n}$ is $2$ sheeted unramified normal covering of $\Gamma_n, ~\forall~ n \geq 1$ with Galois group $\displaystyle \frac{\mathbb{Z}}{2\mathbb{Z}}.$ In fact, for any $n > 1, r \geq 1$ the graph $\Gamma_{n+r}$ is $2n$ sheeted unramified, non normal covering of $\Gamma_r.$ In order to do this we give the definition of the $generalized$ $replacement$ $product$ of Schreier graphs. We also show the corresponding results in zig zag product of Schreier graphs $\Gamma_n$ with a $4$ cycle.