Invariant measures and growth conditions

Let G be a finitely-generated group acting on a set X and let A be a nonempty subset of X . If G has polynomial growth then there exists a finitely-additive G -invariant positive extended real-valued measure μ μ defined on all subsets of X such that μ ( A ) = 1 μ (A) = 1 . When G is solvable, it has polynomial growth if and only if it does not contain a free subsemigroup on two generators. If G contains a free subsemigroup S on two generators, then G has exponential growth and there does not exist a measure μ μ as above with G acting on itself by multiplication and A = S A = S .