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doi: 10.3934/dcds.2020350

Measures and stabilizers of group Cantor actions

1. 

Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria

2. 

Faculty of Mathematics and Computer Science, Jagiellonian University in Krakow, ul. Łojasiewicza 6, 30-348 Kraków, Poland

* Corresponding author

Received  April 2020 Revised  August 2020 Published  October 2020

Fund Project: The first author is supported by DFG grant GR 4899/1-1. The second author is supported by FWF Project P31950-N35

We consider a minimal equicontinuous action of a finitely generated group $ G $ on a Cantor set $ X $ with invariant probability measure $ \mu $, and the stabilizers of points for such an action. We give sufficient conditions under which there exists a subgroup $ H $ of $ G $ such that the set of points in $ X $ whose stabilizers are conjugate to $ H $ has full measure. The conditions are that the action is locally quasi-analytic and locally non-degenerate. An action is locally quasi-analytic if its elements have unique extensions on subsets of uniform diameter. The condition that the action is locally non-degenerate is introduced in this paper. We apply our results to study the properties of invariant random subgroups induced by minimal equicontinuous actions on Cantor sets and to certain almost one-to-one extensions of equicontinuous actions.

Citation: Maik Gröger, Olga Lukina. Measures and stabilizers of group Cantor actions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020350
References:
[1]

M. Abért and G. Elek, Non-abelian free groups admit non-essentially free actions on rooted trees, preprint, arXiv: 0707.0970. Google Scholar

[2]

M. AbértY. Glasner and B. Virág, Kesten's theorem for invariant random subgroups, Duke Math. J., 163 (2014), 465-488.  doi: 10.1215/00127094-2410064.  Google Scholar

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F. Bencs and L. M. Tóth, Invariant random subgroups of groups acting on rooted trees, preprint, arXiv: 1801.05801. Google Scholar

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M. G. BenliR. Grigorchuk and T. Nagnibeda, Universal groups of intermediate growth and their invariant random subgroups, Funct. Anal. Appl., 49 (2015), 159-174.  doi: 10.1007/s10688-015-0101-4.  Google Scholar

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N. Bergeron and D. Gaboriau, Asymptotique des nombres de Betti, invariants $l^2$ et laminations, Comment. Math. Helv., 79 (2004), 362-395.  doi: 10.1007/s00014-003-0798-1.  Google Scholar

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L. Bowen, Random walks on random coset spaces with applications to Furstenberg entropy, Invent. Math., 196 (2014), 485-510.  doi: 10.1007/s00222-013-0473-0.  Google Scholar

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L. Bowen, Invariant random subgroups of the free group, Groups Geom. Dyn., 9 (2015), 891-916.  doi: 10.4171/GGD/331.  Google Scholar

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L. BowenR. Grigorchuk and R. Kravchenko, Invariant random subgroups of lamplighter groups, Israel J. Math., 207 (2015), 763-782.  doi: 10.1007/s11856-015-1160-1.  Google Scholar

[12]

M. I. Cortez and K. Medynets, Orbit equivalence rigidity of equicontinuous systems, J. Lond. Math. Soc. (2), 94 (2016), 545-556.  doi: 10.1112/jlms/jdw047.  Google Scholar

[13]

M. I. Cortez and S. Petite, $G$-odometers and their almost one-to-one extensions, J. Lond. Math. Soc. (2), 78 (2008), 1-20.  doi: 10.1112/jlms/jdn002.  Google Scholar

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J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, Vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

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A. Dudko and K. Medynets, On invariant random subgroups of block-diagonal limits of symmetric groups, Proc. Amer. Math. Soc., 147 (2019), 2481-2494.  doi: 10.1090/proc/14323.  Google Scholar

[16]

J. DyerS. Hurder and O. Lukina, The discriminant invariant of Cantor group actions, Topology Appl., 208 (2016), 64-92.  doi: 10.1016/j.topol.2016.05.005.  Google Scholar

[17]

J. DyerS. Hurder and O. Lukina, Molino theory for matchbox manifolds, Pacific J. Math., 289 (2017), 91-151.  doi: 10.2140/pjm.2017.289.91.  Google Scholar

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D. B. A. EpsteinK. C. Millett and D. Tischler, Leaves without holonomy, J. London Math. Soc., 16 (1977), 548-552.  doi: 10.1112/jlms/s2-16.3.548.  Google Scholar

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R. Fokkink and L. Oversteegen, Homogeneous weak solenoids, Trans. Amer. Math. Soc., 354 (2002), 3743-3755.  doi: 10.1090/S0002-9947-02-03017-9.  Google Scholar

[21]

T. Gelander, A lecture on invariant random subgroups, in New Directions in Locally Compact Groups, 186-204, London Math. Soc. Lecture Note Ser., 447, Cambridge Univ. Press, Cambridge, 2018.  Google Scholar

[22]

T. Gelander, A view on invariant random subgroups, Proc. Int. Cong. of Math., 1 (2018), 1317-1340.   Google Scholar

[23]

E. Glasner and B. Weiss, Uniformly recurrent subgroups, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., Amer. Math. Soc., Providence, RI, 631 (2015), 63–75. doi: 10.1090/conm/631/12596.  Google Scholar

[24]

R. I. Grigorchuk, Some topics in the dynamics of group actions on rooted trees, Proc. Steklov Inst. Math., 273 (2011), 64-175.  doi: 10.1134/S0081543811040067.  Google Scholar

[25]

R. Grigorchuk, V. Nekrashevych and Z. Šunić, From self-similar groups to self-similar sets and spectra, in Fractal Geometry and Stochastics V, Progr. Probab., Birkhäuser/Springer, Cham, 70 (2015), 175–207. doi: 10.1007/978-3-319-18660-3_11.  Google Scholar

[26]

A. Haefliger, Pseudogroups of local isometries, in Differential Geometry (Santiago de Compostela, 1984) (ed. L. A. Cordero), Res. Notes in Math., Pitman, Boston, MA, 131 (1985), 174–197.  Google Scholar

[27]

S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. Math., 126 (1987), 221-275.  doi: 10.2307/1971401.  Google Scholar

[28]

S. Hurder and O. Lukina, Limit group invariants for wild Cantor actions, to appear in Ergodic Theory Dynam. Systems, arXiv: 1904.11072. Google Scholar

[29]

S. Hurder and O. Lukina, Orbit equivalence and classification of weak solenoids, to appear in Indiana Univ. Math. J., arXiv: 1803.02098. Google Scholar

[30]

S. Hurder and O. Lukina, Wild solenoids, Trans. Amer. Math. Soc., 371 (2019), 4493-4533.  doi: 10.1090/tran/7339.  Google Scholar

[31]

M. KambitesP. V. Silva and B. Steinberg, The spectra of lamplighter groups and Cayley machines, Geom. Dedicata, 120 (2006), 193-227.  doi: 10.1007/s10711-006-9086-8.  Google Scholar

[32]

B. Miller, The existence of measures of a given cocycle, I: Atomless, ergodic $\sigma$-finite measures, Ergodic Theory Dynam. Systems, 28 (2008), 1599-1613.  doi: 10.1017/S0143385707001113.  Google Scholar

[33]

V. Nekrashevych, Self-Similar Groups, Mathematical Survey and Monographs, 117, Americal Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/117.  Google Scholar

[34]

A. Ramsay, Virtual groups and group actions, Advances in Math., 6 (1971), 253-322.  doi: 10.1016/0001-8708(71)90018-1.  Google Scholar

[35]

G. Stuck and R. J. Zimmer, Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2), 139 (1994), 723-747.  doi: 10.2307/2118577.  Google Scholar

[36]

S. Thomas and R. Tucker-Drob, Invariant random subgroups of strictly diagonal limits of finite symmetric groups, Bull. Lond. Math. Soc., 46 (2014), 1007-1020.  doi: 10.1112/blms/bdu060.  Google Scholar

[37]

S. Thomas and R. Tucker-Drob, Invariant random subgroups of inductive limits of finite alternating groups, J. Algebra, 503 (2018), 474-533.  doi: 10.1016/j.jalgebra.2018.02.012.  Google Scholar

[38]

A. M. Vershik, Nonfree actions of countable groups and their characters, J. Math. Sci. (N.Y.), 174 (2011), 6pp. doi: 10.1007/s10958-011-0273-2.  Google Scholar

[39]

A. M. Vershik, Totally nonfree actions and the infinite symmetric group, Moscow Math. J., 12 (2012), 193-212.  doi: 10.17323/1609-4514-2012-12-1-193-212.  Google Scholar

[40]

Y. Vorobets, Notes on the Schreier graphs of the Grigorchuk group, in Dynamical Systems and Group Actions, Contemp. Math., Amer. Math. Soc., Providence, RI, 567 (2012), 221–248. doi: 10.1090/conm/567/11250.  Google Scholar

[41]

T. Zheng, On rigid stabilizers and invariant random subgroups of groups of homeomorphisms, preprint, arXiv: 1901.04428. Google Scholar

show all references

References:
[1]

M. Abért and G. Elek, Non-abelian free groups admit non-essentially free actions on rooted trees, preprint, arXiv: 0707.0970. Google Scholar

[2]

M. AbértY. Glasner and B. Virág, Kesten's theorem for invariant random subgroups, Duke Math. J., 163 (2014), 465-488.  doi: 10.1215/00127094-2410064.  Google Scholar

[3]

J. A. Álvarez López and A. Candel, Equicontinuous foliated spaces, Math. Z., 263 (2009), 725-774.  doi: 10.1007/s00209-008-0432-4.  Google Scholar

[4]

J. Auslander, Minimal Flows and their Extensions, North-Holland Mathematics Studies, Vol. 153, North-Holland Publishing Co., Amsterdam, 1988.  Google Scholar

[5]

L. Auslander and C. C. Moore, Unitary representations of solvable Lie groups, Mem. Amer. Math. Soc., 62 (1966), 199 pp.  Google Scholar

[6]

F. Bencs and L. M. Tóth, Invariant random subgroups of groups acting on rooted trees, preprint, arXiv: 1801.05801. Google Scholar

[7]

M. G. BenliR. Grigorchuk and T. Nagnibeda, Universal groups of intermediate growth and their invariant random subgroups, Funct. Anal. Appl., 49 (2015), 159-174.  doi: 10.1007/s10688-015-0101-4.  Google Scholar

[8]

N. Bergeron and D. Gaboriau, Asymptotique des nombres de Betti, invariants $l^2$ et laminations, Comment. Math. Helv., 79 (2004), 362-395.  doi: 10.1007/s00014-003-0798-1.  Google Scholar

[9]

L. Bowen, Random walks on random coset spaces with applications to Furstenberg entropy, Invent. Math., 196 (2014), 485-510.  doi: 10.1007/s00222-013-0473-0.  Google Scholar

[10]

L. Bowen, Invariant random subgroups of the free group, Groups Geom. Dyn., 9 (2015), 891-916.  doi: 10.4171/GGD/331.  Google Scholar

[11]

L. BowenR. Grigorchuk and R. Kravchenko, Invariant random subgroups of lamplighter groups, Israel J. Math., 207 (2015), 763-782.  doi: 10.1007/s11856-015-1160-1.  Google Scholar

[12]

M. I. Cortez and K. Medynets, Orbit equivalence rigidity of equicontinuous systems, J. Lond. Math. Soc. (2), 94 (2016), 545-556.  doi: 10.1112/jlms/jdw047.  Google Scholar

[13]

M. I. Cortez and S. Petite, $G$-odometers and their almost one-to-one extensions, J. Lond. Math. Soc. (2), 78 (2008), 1-20.  doi: 10.1112/jlms/jdn002.  Google Scholar

[14]

J. de Vries, Elements of Topological Dynamics, Mathematics and its Applications, Vol. 257, Kluwer Academic Publishers Group, Dordrecht, 1993. doi: 10.1007/978-94-015-8171-4.  Google Scholar

[15]

A. Dudko and K. Medynets, On invariant random subgroups of block-diagonal limits of symmetric groups, Proc. Amer. Math. Soc., 147 (2019), 2481-2494.  doi: 10.1090/proc/14323.  Google Scholar

[16]

J. DyerS. Hurder and O. Lukina, The discriminant invariant of Cantor group actions, Topology Appl., 208 (2016), 64-92.  doi: 10.1016/j.topol.2016.05.005.  Google Scholar

[17]

J. DyerS. Hurder and O. Lukina, Molino theory for matchbox manifolds, Pacific J. Math., 289 (2017), 91-151.  doi: 10.2140/pjm.2017.289.91.  Google Scholar

[18]

R. Ellis, Lectures on Topological Dynamics, W. A. Benjamin, Inc., New York, 1969.  Google Scholar

[19]

D. B. A. EpsteinK. C. Millett and D. Tischler, Leaves without holonomy, J. London Math. Soc., 16 (1977), 548-552.  doi: 10.1112/jlms/s2-16.3.548.  Google Scholar

[20]

R. Fokkink and L. Oversteegen, Homogeneous weak solenoids, Trans. Amer. Math. Soc., 354 (2002), 3743-3755.  doi: 10.1090/S0002-9947-02-03017-9.  Google Scholar

[21]

T. Gelander, A lecture on invariant random subgroups, in New Directions in Locally Compact Groups, 186-204, London Math. Soc. Lecture Note Ser., 447, Cambridge Univ. Press, Cambridge, 2018.  Google Scholar

[22]

T. Gelander, A view on invariant random subgroups, Proc. Int. Cong. of Math., 1 (2018), 1317-1340.   Google Scholar

[23]

E. Glasner and B. Weiss, Uniformly recurrent subgroups, in Recent Trends in Ergodic Theory and Dynamical Systems, Contemp. Math., Amer. Math. Soc., Providence, RI, 631 (2015), 63–75. doi: 10.1090/conm/631/12596.  Google Scholar

[24]

R. I. Grigorchuk, Some topics in the dynamics of group actions on rooted trees, Proc. Steklov Inst. Math., 273 (2011), 64-175.  doi: 10.1134/S0081543811040067.  Google Scholar

[25]

R. Grigorchuk, V. Nekrashevych and Z. Šunić, From self-similar groups to self-similar sets and spectra, in Fractal Geometry and Stochastics V, Progr. Probab., Birkhäuser/Springer, Cham, 70 (2015), 175–207. doi: 10.1007/978-3-319-18660-3_11.  Google Scholar

[26]

A. Haefliger, Pseudogroups of local isometries, in Differential Geometry (Santiago de Compostela, 1984) (ed. L. A. Cordero), Res. Notes in Math., Pitman, Boston, MA, 131 (1985), 174–197.  Google Scholar

[27]

S. Hurder and A. Katok, Ergodic theory and Weil measures for foliations, Ann. Math., 126 (1987), 221-275.  doi: 10.2307/1971401.  Google Scholar

[28]

S. Hurder and O. Lukina, Limit group invariants for wild Cantor actions, to appear in Ergodic Theory Dynam. Systems, arXiv: 1904.11072. Google Scholar

[29]

S. Hurder and O. Lukina, Orbit equivalence and classification of weak solenoids, to appear in Indiana Univ. Math. J., arXiv: 1803.02098. Google Scholar

[30]

S. Hurder and O. Lukina, Wild solenoids, Trans. Amer. Math. Soc., 371 (2019), 4493-4533.  doi: 10.1090/tran/7339.  Google Scholar

[31]

M. KambitesP. V. Silva and B. Steinberg, The spectra of lamplighter groups and Cayley machines, Geom. Dedicata, 120 (2006), 193-227.  doi: 10.1007/s10711-006-9086-8.  Google Scholar

[32]

B. Miller, The existence of measures of a given cocycle, I: Atomless, ergodic $\sigma$-finite measures, Ergodic Theory Dynam. Systems, 28 (2008), 1599-1613.  doi: 10.1017/S0143385707001113.  Google Scholar

[33]

V. Nekrashevych, Self-Similar Groups, Mathematical Survey and Monographs, 117, Americal Mathematical Society, Providence, RI, 2005. doi: 10.1090/surv/117.  Google Scholar

[34]

A. Ramsay, Virtual groups and group actions, Advances in Math., 6 (1971), 253-322.  doi: 10.1016/0001-8708(71)90018-1.  Google Scholar

[35]

G. Stuck and R. J. Zimmer, Stabilizers for ergodic actions of higher rank semisimple groups, Ann. of Math. (2), 139 (1994), 723-747.  doi: 10.2307/2118577.  Google Scholar

[36]

S. Thomas and R. Tucker-Drob, Invariant random subgroups of strictly diagonal limits of finite symmetric groups, Bull. Lond. Math. Soc., 46 (2014), 1007-1020.  doi: 10.1112/blms/bdu060.  Google Scholar

[37]

S. Thomas and R. Tucker-Drob, Invariant random subgroups of inductive limits of finite alternating groups, J. Algebra, 503 (2018), 474-533.  doi: 10.1016/j.jalgebra.2018.02.012.  Google Scholar

[38]

A. M. Vershik, Nonfree actions of countable groups and their characters, J. Math. Sci. (N.Y.), 174 (2011), 6pp. doi: 10.1007/s10958-011-0273-2.  Google Scholar

[39]

A. M. Vershik, Totally nonfree actions and the infinite symmetric group, Moscow Math. J., 12 (2012), 193-212.  doi: 10.17323/1609-4514-2012-12-1-193-212.  Google Scholar

[40]

Y. Vorobets, Notes on the Schreier graphs of the Grigorchuk group, in Dynamical Systems and Group Actions, Contemp. Math., Amer. Math. Soc., Providence, RI, 567 (2012), 221–248. doi: 10.1090/conm/567/11250.  Google Scholar

[41]

T. Zheng, On rigid stabilizers and invariant random subgroups of groups of homeomorphisms, preprint, arXiv: 1901.04428. Google Scholar

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