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Dynamical properties of profinite actions

Published online by Cambridge University Press:  16 November 2011

MIKLÓS ABÉRT  and
GÁBOR ELEK
MIKLÓS ABÉRT
Affiliation:
Alfred Renyi Institute of the Hungarian Academy of Sciences, 13–15 Realtanoda u., 1053 Budapest, Hungary (email: elek@renyi.hu)
GÁBOR ELEK
Affiliation:
Alfred Renyi Institute of the Hungarian Academy of Sciences, 13–15 Realtanoda u., 1053 Budapest, Hungary (email: elek@renyi.hu)
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Abstract

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky’s property (τ) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to the question of Lubotzky and Zuk: for families of subgroups, is property (τ) inherited by the lattice of subgroups generated by the family? On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicit estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expanding covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 32 , Issue 6 , December 2012 , pp. 1805 - 1835
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Abért, M. and Nikolov, N.. Rank gradient, cost of groups and the rank versus Heegard genus problem. Preprint.Google Scholar
[2]Abért, M., Jaikin-Zapirain, A. and Nikolov, N.. The rank gradient from a combinatorial viewpoint. Groups Geom. Dyn. 5 (2011), 213–230.CrossRefGoogle Scholar
[3]Abért, M.. On chains of subgroups in residually finite groups. Preprint.Google Scholar
[4]Abért, M. and Virág, B.. Dimension and randomness in groups acting on rooted trees. J. Amer. Math. Soc. 18(1) (2005), 157–192.CrossRefGoogle Scholar
[5]Bollobás, B.. The independence ratio of regular graphs. Proc. Amer. Math. Soc. 83(2) (1981), 433–436.CrossRefGoogle Scholar
[6]Bowen, L.. Periodicity and circle packings of the hyperbolic plane. Geom. Dedicata 102 (2003), 213–236.CrossRefGoogle Scholar
[7]Conley, C. T. and Kechris, A.. Measurable chromatic and independence numbers for ergodic graphs and group actions. Preprint, 2010.Google Scholar
[8]Connes, A. and Weiss, B.. Property T and asymptotically invariant sequences. Israel J. Math. 37(3) (1980), 209–210.CrossRefGoogle Scholar
[9]Desai, M. and Rao, V.. A characterization of the smallest eigenvalue of a graph. J. Graph Theory 18(2) (1994), 181–194.CrossRefGoogle Scholar
[10]Epstein, I.. Orbit inequivalent actions of non-amenable groups. Preprint.Google Scholar
[11]Foreman, M. and Weiss, B.. An anti-classification theorem for ergodic measure preserving transformations. J. Eur. Math. Soc. (JEMS) 6(3) (2004), 277–292.CrossRefGoogle Scholar
[12]Friedman, J.. Relative expanders or weakly relatively Ramanujan graphs. Duke Math. J. 118(1) (2003), 19–35.CrossRefGoogle Scholar
[13]Glasner, E. and Weiss, B.. Kazhdan’s property T and the geometry of the collection of invariant measures. Geom. Funct. Anal. 7(5) (1997), 917–935.CrossRefGoogle Scholar
[14]Grigorchuk, R. I., Nekrashevich, V. V. and Suschanskii, V. I. Automata, dynamical systems, and groups. Proc. Steklov Inst. Math. 231(4) (2000), 128–203.Google Scholar
[15]Gruenberg, K. W.. Residual properties of infinite soluble groups. Proc. Lond. Math. Soc. 7 (1957), 29–62.CrossRefGoogle Scholar
[16]Hjorth, G. and Kechris, A. S.. Rigidity theorems for actions of product groups and countable Borel equivalence relations. Mem. Amer. Math. Soc. 177(833) (2005).Google Scholar
[17]Ioana, A.. Cocycle superrigidity for profinite actions of Kazhdan groups. Preprint.Google Scholar
[18]Kechris, A.. Global Aspects of Ergodic Group Actions (Mathematical Surveys and Monographs, 160). American Mathematical Society, Providence, RI, 2010.CrossRefGoogle Scholar
[19]Kechris, A. and Miller, B.. Topics in Orbit Equivalence (Lecture Notes in Mathematics, 1852). Springer, Berlin, 2004.CrossRefGoogle Scholar
[20]Losert, V. and Rindler, H.. Almost invariant sets. Bull. Lond. Math. Soc. 13(2) (1981), 145–148.CrossRefGoogle Scholar
[21]Lubotzky, A.. Discrete Groups, Expanding Graphs and Invariant Measures (Progress in Mathematics, 125). Birkhäuser, Basel, 1994.CrossRefGoogle Scholar
[22]Lubotzky, A. and Segal, D.. Subgroup Growth (Progress in Mathematics, 212). Birkhäuser, Basel, 2003.CrossRefGoogle Scholar
[23]Lubotzky, A. and Zuk, A.. Property (τ), preliminary version, 2003, http://www.ma.huji.ac.il/∼alexlub .Google Scholar
[24]Ozawa, N. and Popa, S.. On a class of II_1 factors with at most one Cartan subalgebra. Ann. of Math. (2) 172(1) (2010), 713–749.CrossRefGoogle Scholar
[25]Schmidt, K.. Amenability, Kazhdan’s property T, strong ergodicity and invariant means for ergodic group-actions. Ergod. Th. & Dynam. Sys. 1(2) (1981), 223–236.CrossRefGoogle Scholar
[26]Shalom, Y.. Expanding graphs and invariant means. Combinatorica 17(4) (1997), 555–575.CrossRefGoogle Scholar
[27]Wilson, J.. Profinite Groups (London Mathematical Society Monographs. New Series). Oxford University Press, 1998.CrossRefGoogle Scholar