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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Invariant measures and growth conditions


Author: Joseph Max Rosenblatt
Journal: Trans. Amer. Math. Soc. 193 (1974), 33-53
MSC: Primary 43A07; Secondary 20F15
DOI: https://doi.org/10.1090/S0002-9947-1974-0342955-9
MathSciNet review: 0342955
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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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DOI: https://doi.org/10.1090/S0002-9947-1974-0342955-9
Article copyright: © Copyright 1974 American Mathematical Society