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

Abstract

For every infinite sequence ω=x 1x 2..., with x i∈{0, 1}, we construct an infinite 4-regular graph X ω. These graphs are precisely the Schreier graphs of the action of a certain self-similar group on the space {0, 1} ∞. We solve the isomorphism and local isomorphism problems for these graphs, and determine their automorphism groups. Finally, we prove that all graphs X ω have intermediate growth. © 2012 Elsevier Ltd.