Nonexistence of invariant measures

Promislow, David

doi:10.1090/S0002-9939-1983-0691283-1

Nonexistence of invariant measures
HTML articles powered by AMS MathViewer

by David Promislow PDF

Proc. Amer. Math. Soc. 88 (1983), 89-92 Request permission

Abstract:

Let $G$ be a group acting on a set $X$. Suppose that for some positive integer $r$, $G$ contains a free group $F$ of rank $> r$ and the intersection of any stabilizer with $F$ has rank $\leqslant r$. A graph theoretic approach is used to show that there is no invariant measure on $X$.

References

Charles A. Akemann, Operator algebras associated with Fuchsian groups, Houston J. Math. 7 (1981), no. 3, 295–301. MR 640971
Claude Berge, Graphs and hypergraphs, North-Holland Mathematical Library, Vol. 6, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. Translated from the French by Edward Minieka. MR 0357172
J. Dixmier and E. C. Lance, Deux nouveaux facteurs de type $\textrm {II}_{1}$, Invent. Math. 7 (1969), 226–234 (French). MR 248535, DOI 10.1007/BF01404307
Edward G. Effros, Property $\Gamma$ and inner amenability, Proc. Amer. Math. Soc. 47 (1975), 483–486. MR 355626, DOI 10.1090/S0002-9939-1975-0355626-6
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
Marshall Hall Jr., The theory of groups, The Macmillan Company, New York, N.Y., 1959. MR 0103215
Wilfried Imrich, Subgroup theorems and graphs, Combinatorial mathematics, V (Proc. Fifth Austral. Conf., Roy. Melbourne Inst. Tech., Melbourne, 1976) Lecture Notes in Math., Vol. 622, Springer, Berlin, 1977, pp. 1–27. MR 0463016

Combinatorial group theory

Joseph Max Rosenblatt, A generalization of Følner’s condition, Math. Scand. 33 (1973), 153–170. MR 333068, DOI 10.7146/math.scand.a-11481
Joseph Rosenblatt, Uniqueness of invariant means for measure-preserving transformations, Trans. Amer. Math. Soc. 265 (1981), no. 2, 623–636. MR 610970, DOI 10.1090/S0002-9947-1981-0610970-7