May  2021, 41(5): 2001-2029. 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  May 2021 Early access  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 and Continuous Dynamical Systems, 2021, 41 (5) : 2001-2029. 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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[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.

[33]

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

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

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.

[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.

[3]

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

[4]

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

[5]

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

[6]

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

[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.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[22]

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

[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.

[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.

[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.

[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.

[27]

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

[28]

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

[29]

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

[30]

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

[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.

[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.

[33]

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

[34]

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

[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.

[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.

[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.

[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.

[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.

[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.

[41]

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

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