A necessary and sufficient condition for a connected amenable group to have polynomial growth

Ganesan, S.

doi:10.1090/S0002-9939-1985-0766551-7

A necessary and sufficient condition for a connected amenable group to have polynomial growth
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by S. Ganesan PDF

Proc. Amer. Math. Soc. 93 (1985), 176-178 Request permission

Abstract:

It is shown that a connected amenable group $G$ has polynomial growth if, and only if, given any open subsemigroup $S$ of $G$ and a compact set $K$ in $G$ there exists an $s$ in $S$ such that ${K_s} \subset S$.

References

Jacques Dixmier, Opérateurs de rang fini dans les représentations unitaires, Inst. Hautes Études Sci. Publ. Math. 6 (1960), 13–25 (French). MR 136684
S. Ganesan and J. W. Jenkins, An Archimedian property for groups with polynomial growth, Proc. Amer. Math. Soc. 88 (1983), no. 3, 550–554. MR 699432, DOI 10.1090/S0002-9939-1983-0699432-6
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
A. Hulanicki, Subalgebra of $L_{1}(G)$ associated with Laplacian on a Lie group, Colloq. Math. 31 (1974), 259–287. MR 372536, DOI 10.4064/cm-31-2-259-287
Joe W. Jenkins, Amenable subsemigroups of a locally compact group, Proc. Amer. Math. Soc. 25 (1970), 766–770. MR 263967, DOI 10.1090/S0002-9939-1970-0263967-1
J. W. Jenkins, Følner’s condition for exponentially bounded groups, Math. Scand. 35 (1974), 165–174. MR 369609, DOI 10.7146/math.scand.a-11544
J. W. Jenkins, Representations of exponentially bounded groups, Amer. J. Math. 98 (1976), no. 1, 29–38. MR 454520, DOI 10.2307/2373611
Anthony To-ming Lau, Invariant means on dense subsemigroups of topolgical groups, Canadian J. Math. 23 (1971), 797–801. MR 288209, DOI 10.4153/CJM-1971-088-4