This question is motivated by this one http://mathoverflow.net/questions/32899/what-is-the-relation-between-the-number-syntactic-congruence-classes-and-the-num where it essentially asks to compare the growth of the syntactic monoid with the growth of the minimal automaton. A special case is the following. Let G be an infinite finitely generated group and H a subgroup such that G acts faithfully on G/H. How different can the growth of G and the Schreier graph of G/H be?
I know the Grigrchuk group of intermediate growth has faithful Schreier graphs of polynomial growth.
Are there groups of exponential growth with faithful Schreier graphs of polynomial growth?
Schreier graphs of non-elementary hyperbolic groups with respect to infinite index quasi-convex subgroups have non-amenable Schreier graphs so ths should be avoided.