Zeta functions of finite Schreier graphs and their zig zag products

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 ${\Gamma }_n,\forall n\ge 1$ with Galois group $\frac{\mathbb{Z}}{2\mathbb{Z}}.$ In fact, for any $n>1,r\ge 1$ the graph ${\Gamma }_{n+r}$ is $2^n$ 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.

30 pages, 6 figures, 13 tables, Preprint of an article submitted for consideration in Journal of Algebra and Its Applications ©2016, copyright WorldScientific Publishing Company, http://www.worldscientific.com/worldscinet/jaa