Topological dynamics and group theory

Glasner, Shmuel

doi:10.1090/S0002-9947-1974-0336723-1

Topological dynamics and group theory
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Trans. Amer. Math. Soc. 187 (1974), 327-334 Request permission

Abstract:

We prove, using notions and techniques of topological dynamics, that a nonamenable group contains a finitely-generated subgroup of exponential growth. We also show that a group which belongs to a certain class, defined by means of topological dynamical properties, always contains a free subgroup on two generators.

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
E. B. Dynkin and M. B. Maljutov, Random walk on groups with a finite number of generators, Dokl. Akad. Nauk SSSR 137 (1961), 1042–1045 (Russian). MR 0131904
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
Harry Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49. MR 213508, DOI 10.1007/BF01692494

Random walks and discrete sub-groups of Lie groups

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Harry Furstenberg, A Poisson formula for semi-simple Lie groups, Ann. of Math. (2) 77 (1963), 335–386. MR 146298, DOI 10.2307/1970220
Shmuel Glasner, Compressibility properties in topological dynamics, Amer. J. Math. 97 (1975), 148–171. MR 365537, DOI 10.2307/2373665
Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. MR 112053, DOI 10.7146/math.scand.a-10568
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
Joseph A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Differential Geometry 2 (1968), 421–446. MR 248688