P-amenable locally compact groups

LetG denote a locally compact group andP(G) P _c (G)] the topological semigroup of absolutely continuous absolutely continuous and compactly supported] probability measures onG. We say thatG isP-amenable P _c -amenable] if the topological semigroupP(G)P _c (G)] is amenable. Some combinatorial properties of this class of groups are studied. The relationship between amenability andP-amenability ofG is investigated. It is shown that for a connected solvableP _c -amenable groupG, G has polynomial growth if, and only if,P _c (S)={fP _c (G)suppfS} is amenable for any open subsemigroupS ofG.This paper contains a part of the author's doctoral thesis at the State University of New York at Albany. The author wishes to thank ProfessorJoe Jenkins for his valuable suggestions and encouragement during the course of this work.