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

For every infinite sequence ω = x 1 x 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.