On a family of Schreier graphs of intermediate growth associated with a self-similar group

Abstract: For every infinite sequence $\omega=x_1,x_2,...$, with $x_i\in{0,1}$, we construct an infinite 4-regular graph $X_{\omega}$. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space ${0,1}^{\infty}$. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs $X_\omega$ have intermediate growth.