Invariant measures and growth conditions

Rosenblatt, Joseph Max

doi:10.1090/S0002-9947-1974-0342955-9

Invariant measures and growth conditions
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Trans. Amer. Math. Soc. 193 (1974), 33-53 Request permission

Abstract:

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 $\mu$ defined on all subsets of X such that $\mu (A) = 1$. 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 $\mu$ as above with G acting on itself by multiplication and $A = S$.

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