Invariant measures and growth conditions

Rosenblatt, Joseph Max

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

Invariant measures and growth conditions
HTML articles powered by AMS MathViewer

by Joseph Max Rosenblatt

Trans. Amer. Math. Soc. 193 (1974), 33-53

DOI: https://doi.org/10.1090/S0002-9947-1974-0342955-9

PDF | 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 $A = S$.

References

H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. (3) 25 (1972), 603–614. MR 379672, DOI 10.1112/plms/s3-25.4.603
A. I. Borevich and I. R. Shafarevich, Number theory, Pure and Applied Mathematics, Vol. 20, Academic Press, New York-London, 1966. Translated from the Russian by Newcomb Greenleaf. MR 0195803
W. R. Emerson and F. P. Greenleaf, Asymptotic behavior of products $C^{p}=C+\cdots +C$ in locally compact abelian groups, Trans. Amer. Math. Soc. 145 (1969), 171–204. MR 249535, DOI 10.1090/S0002-9947-1969-0249535-2

Studies in amenable semigroups

Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto-London, 1969. MR 0251549
M. Hochster, Subsemigroups of amenable groups, Proc. Amer. Math. Soc. 21 (1969), 363–364. MR 240223, DOI 10.1090/S0002-9939-1969-0240223-0
Kenneth Hoffman and Ray Kunze, Linear algebra, Prentice-Hall Mathematics Series, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1961. MR 0125849
A. Hulanicki, On symmetry of group algebras of discrete nilpotent groups, Studia Math. 35 (1970), 207–219 (errata insert). MR 278082, DOI 10.4064/sm-35-2-207-219
A. Hulanicki, On positive functionals on a group algebra multiplicative on a subalgebra, Studia Math. 37 (1970/71), 163–171. MR 310547, DOI 10.4064/sm-37-2-163-171
Joe W. Jenkins, On the spectral radius of elements in a group algebra, Illinois J. Math. 15 (1971), 551–554. MR 287332
J. W. Jenkins, Growth of connected locally compact groups, J. Functional Analysis 12 (1973), 113–127. MR 0349895, DOI 10.1016/0022-1236(73)90092-x
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 0207802
A. I. Mal′cev, On some classes of infinite soluble groups, Mat. Sbornik N.S. 28(70) (1951), 567–588 (Russian). MR 0043088
J. Milnor, A note on curvature and fundamental group, J. Differential Geometry 2 (1968), 1–7. MR 232311, DOI 10.4310/jdg/1214501132
John Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2 (1968), 447–449. MR 244899

Groups with the fixed point property

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 J. Rotman, The theory of groups. An introduction, Allyn and Bacon, Inc., Boston, Mass., 1965. MR 0204499

Zur Allgemeinen Theorie der Masses

13

Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688
S. Balcerzyk and Jan Mycielski, On the existence of free subgroups in topological groups, Fund. Math. 44 (1957), 303–308. MR 94417, DOI 10.4064/fm-44-3-303-308