# Rigid actions of amenable groups

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## Abstract

Given a countable discrete amenable group *G*, does there exist a free action of *G* on a Lebesgue probability space which is both rigid and weakly mixing? The answer to this question is positive if *G* is abelian. An affirmative answer is given in this paper, in the case that *G* is solvable or residually finite. For a locally finite group, the question is reduced to an algebraic one. It is exemplified how the algebraic question can be positively resolved for some groups, whereas for others the algebraic viewpoint suggests the answer may be negative.

## Keywords

Normal Subgroup Measure Preserve Cayley Graph Free Action Amenable Group
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## References

- [1]V. V. Belyaev,
*Topologization of countable locally finite groups*, Algebra i Logika**34**(1995), 613–618, 728.zbMATHMathSciNetGoogle Scholar - [2]V. Bergelson and J. Rosenblatt,
*Mixing actions of groups*, Illinois Journal of Mathematics**32**(1988), 65–80.zbMATHMathSciNetGoogle Scholar - [3]J. Bourgain and A. Gamburd,
*New results on expanders*, Comptes Rendus Mathématique. Académie des Sciences. Paris**342**(2006), 717–721.zbMATHMathSciNetGoogle Scholar - [4]H. A. Dye,
*On the ergodic mixing theorem*, Transactions of the American Mathematical Society**118**(1965), 123–130.zbMATHCrossRefMathSciNetGoogle Scholar - [5]M. Foreman and B. Weiss,
*An anti-classification theorem for ergodic measure preserving transformations*, Journal of the European Mathematical Society (JEMS)**6**(2004), 277–292.zbMATHMathSciNetCrossRefGoogle Scholar - [6]H. Furstenberg and B. Weiss,
*The finite multipliers of infinite ergodic transformations*, in*The Structure of Attractors in Dynamical Systems (Proc. Conf., North Dakota State 90 IDDO SAMET Isr. J. Math. Univ., Fargo, N.D., 1977)*, Lecture Notes in Math., vol. 668, Springer, Berlin, 1978, pp. 127–132.Google Scholar - [7]E. Glasner and J. L. King,
*A zero-one law for dynamical properties*, in*Topological Dynamics and Applications (Minneapolis, MN, 1995)*, Contemporary Mathematics, vol. 215, American Mathematical Society, Providence, RI, 1998, pp. 231–242.Google Scholar - [8]E. Glasner, J.-P. Thouvenot and B. Weiss,
*Every countable group has the weak rohlin property*, Proceedings of the London Mathematical Society, to appear.Google Scholar - [9]E. Glasner and B. Weiss,
*Minimal actions of the group S(Z) of permutations of the integers*, Geometric and Functional Analysis**12**(2002), 964–988.zbMATHCrossRefMathSciNetGoogle Scholar - [10]F. P. Greenleaf,
*Invariant Means on Topological Groups and their Applications*, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York, 1969.zbMATHGoogle Scholar - [11]P. Hall,
*Some constructions for locally finite groups*, Journal of the London Mathematical Society. Second Series 34 (1959), 305–319.zbMATHMathSciNetGoogle Scholar - [12]P. R. Halmos,
*Lectures on Ergodic Theory*, Chelsea Publishing Co., New York, 1960.zbMATHGoogle Scholar - [13]A. A. Klyachko and A. V. Trofimov,
*The number of non-solutions of an equation in a group*, Journal of Group Theory**8**(2005), 747–754.zbMATHCrossRefMathSciNetGoogle Scholar - [14]A. Lubotzky,
*Discrete Groups, Expanding Graphs and Invariant Measures*, Progress in Mathematics, vol. 125, Birkhäauser Verlag, Basel, 1994.zbMATHGoogle Scholar - [15]A. A. Markov,
*Three papers on topological groups: I. On the existence of periodic connected topological groups. II. On free topological groups. III. On unconditionally closed sets*, American Mathematical Society Translations**1950**(1950), 120.MathSciNetGoogle Scholar - [16]D. S. Ornstein and B. Weiss,
*Entropy and isomorphism theorems for actions of amenable groups*, Journal d’Analyse Mathématique**48**(1987), 1–141.zbMATHCrossRefMathSciNetGoogle Scholar - [17]D. J. Rudolph,
*Fundamentals of Measurable Dynamics*, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1990.zbMATHGoogle Scholar - [18]K. Schmidt and P. Walters,
*Mildly mixing actions of locally compact groups*, Proceedings of the London Mathematical Society. Third Series**45**(1982), 506–518.Google Scholar - [19]M. Suzuki,
*Group Theory. I*, vol. 247, Springer-Verlag, Berlin, 1982.Google Scholar - [20]P. Walters,
*Some invariant σ-algebras for measure-preserving transformations*, Transactions of the American Mathematical Society**163**(1972), 357–368.zbMATHCrossRefMathSciNetGoogle Scholar

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