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Mosc. Math. J., 2012, Volume 12, Number 1, Pages 193–212 (Mi mmj453)  

This article is cited in 29 scientific papers (total in 30 papers)

Totally nonfree actions and the infinite symmetric group

A. M. Vershikab

a St. Petersburg Department of Steklov Institute of Mathematics
b Max Planck Institute, Bonn

Abstract: We consider totally nonfree (TNF) actions of groups and the corresponding adjoint invariant (AD) measures on lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF measures on the lattice of subgroups of the infinite symmetric group $S_\mathbb N$. The problem is closely related to the theory of characters and factor representations of groups.

Key words and phrases: totally nonfree actions, infinite symmetric group, random subgroups.

DOI: https://doi.org/10.17323/1609-4514-2012-12-1-193-212

Full text: http://www.ams.org/.../abst12-1-2012.html
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MSC: 37A15, 20B35, 22D40
Received: October 4, 2011
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Citation: A. M. Vershik, “Totally nonfree actions and the infinite symmetric group”, Mosc. Math. J., 12:1 (2012), 193–212

Citation in format AMSBIB
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\paper Totally nonfree actions and the infinite symmetric group
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\yr 2012
\vol 12
\issue 1
\pages 193--212
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Yu. A. Neretin, “A remark on representations of infinite symmetric groups”, J. Math. Sci. (N. Y.), 190:3 (2013), 464–467  mathnet  crossref  mathscinet
    2. V. M. Buchstaber, M. I. Gordin, I. A. Ibragimov, V. A. Kaimanovich, A. A. Kirillov, A. A. Lodkin, S. P. Novikov, A. Yu. Okounkov, G. I. Olshanski, F. V. Petrov, Ya. G. Sinai, L. D. Faddeev, S. V. Fomin, N. V. Tsilevich, Yu. V. Yakubovich, “Anatolii Moiseevich Vershik (on his 80th birthday)”, Russian Math. Surveys, 69:1 (2014), 165–179  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    3. Gorin V., Kerov S., Vershik A., “Finite Traces and Representations of the Group of Infinite Matrices Over a Finite Field”, Adv. Math., 254 (2014), 331–395  crossref  mathscinet  zmath  isi  elib  scopus
    4. Bowen L., “Random Walks on Random Coset Spaces with Applications to Furstenberg Entropy”, Invent. Math., 196:2 (2014), 485–510  crossref  mathscinet  zmath  isi  elib  scopus
    5. Abert M., Glasner Ya., Virag B., “Kesten's Theorem for Invariant Random Subgroups”, Duke Math. J., 163:3 (2014), 465–488  crossref  mathscinet  zmath  isi  elib  scopus
    6. Grigorchuk R., Savchuk D., “Self-Similar Groups Acting Essentially Freely on the Boundary of the Binary Rooted Tree”, Group Theory, Combinatorics, and Computing, Contemporary Mathematics, 611, eds. Morse R., NikolovaPopova D., Witherspoon S., Amer Mathematical Soc, 2014, 9–48  crossref  mathscinet  zmath  isi
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    9. Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Russian Math. Surveys, 70:4 (2015), 715–773  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    10. M. G. Benli, R. I. Grigorchuk, T. V. Nagnibeda, “Universal Groups of Intermediate Growth and Their Invariant Random Subgroups”, Funct. Anal. Appl., 49:3 (2015), 159–174  mathnet  crossref  crossref  isi  elib
    11. A. M. Vershik, N. I. Nessonov, “Stable representations of the infinite symmetric group”, Izv. Math., 79:6 (2015), 1184–1214  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    12. Bowen L., Grigorchuk R., Kravchenko R., “Invariant Random Subgroups of Lamplighter Groups”, Isr. J. Math., 207:2 (2015), 763–782  crossref  mathscinet  zmath  isi  elib  scopus
    13. D'Angeli D., Rodaro E., “a Geometric Approach To (Semi)-Groups Defined By Automata Via Dual Transducers”, Geod. Dedic., 174:1 (2015), 375–400  crossref  mathscinet  zmath  isi  scopus
    14. Cannizzo J., “the Boundary Action of a Sofic Random Subgroup of the Free Group”, Group. Geom. Dyn., 9:3 (2015), 683–709  crossref  zmath  isi  elib  scopus
    15. Bowen L., “Invariant Random Subgroups of the Free Group”, Group. Geom. Dyn., 9:3 (2015), 891–916  crossref  zmath  isi  elib  scopus
    16. Glasner E., Weiss B., “Uniformly Recurrent Subgroups”, Recent Trends in Ergodic Theory and Dynamical Systems, Contemporary Mathematics, 631, eds. Bhattacharya S., Das T., Ghosh A., Shah R., Amer Mathematical Soc, 2015, 63–75  crossref  mathscinet  zmath  isi
    17. D. D'Angeli, E. Rodaro, “Freeness of automaton groups vs boundary dynamics”, J. Algebra, 462 (2016), 115–136  crossref  mathscinet  zmath  isi  scopus
    18. A. M. Vershik, “Asymptotic theory of path spaces of graded graphs and its applications”, Jap. J. Math., 11:2 (2016), 151–218  crossref  mathscinet  zmath  isi  scopus
    19. D. Creutz, “Stabilizers of actions of lattices in products of groups”, Ergod. Theory Dyn. Syst., 37:4 (2017), 1133–1186  crossref  mathscinet  zmath  isi  scopus
    20. I. Biringer, O. Tamuz, “Unimodularity of invariant random subgroups”, Trans. Am. Math. Soc., 369:6 (2017), 4043–4061  crossref  mathscinet  zmath  isi  scopus
    21. D. Creutz, J. Peterson, “Stabilizers of ergodic actions of lattices and commensurators”, Trans. Am. Math. Soc., 369:6 (2017), 4119–4166  crossref  mathscinet  zmath  isi  scopus
    22. Ya. Glasner, “Invariant random subgroups of linear groups”, Isr. J. Math., 219:1 (2017), 215–270  crossref  mathscinet  zmath  isi  scopus
    23. L. Bowen, R. Grigorchuk, R. Kravchenko, “Characteristic random subgroups of geometric groups and free abelian groups of infinite rank”, Trans. Am. Math. Soc., 369:2 (2017), 755–781  crossref  mathscinet  zmath  isi  scopus
    24. I. Biringer, J. Raimbault, “Ends of unimodular random manifolds”, Proc. Amer. Math. Soc., 145:9 (2017), 4021–4029  crossref  mathscinet  zmath  isi  scopus
    25. M. Abert, N. Bergeron, I. Biringer, Ts. Gelander, N. Nikolov, J. Raimbault, I. Samet, “On the growth of $L^2$-invariants for sequences of lattices in Lie groups”, Ann. Math., 185:3 (2017), 711–790  crossref  mathscinet  zmath  isi  scopus
    26. Hartman Ya., Yadin A., “Furstenberg Entropy of Intersectional Invariant Random Subgroups”, Compos. Math., 154:10 (2018), 2239–2265  crossref  mathscinet  zmath  isi  scopus
    27. D'Angeli D., Rodaro E., “Fragile Words and Cayley Type Transducers”, Int. J. Group Theory, 7:3, 3 (2018), 95–109  crossref  mathscinet  isi
    28. Dudko A., Grigorchuk R., “On Diagonal Actions of Branch Groups and the Corresponding Characters”, J. Funct. Anal., 274:11 (2018), 3033–3055  crossref  mathscinet  zmath  isi  scopus
    29. Thomas S., Tucker-Drob R., “Invariant Random Subgroups of Inductive Limits of Finite Alternating Groups”, J. Algebra, 503 (2018), 474–533  crossref  mathscinet  zmath  isi  scopus
    30. Elek G., “Uniformly Recurrent Subgroups and Simple C-Algebras”, J. Funct. Anal., 274:6 (2018), 1657–1689  crossref  mathscinet  zmath  isi  scopus
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