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Tr. Mat. Inst. Steklova, 2011,	Volume 273,	Pages 72–191	(Mi tm3283)
This article is cited in 36 scientific papers (total in 36 papers)
Some topics in the dynamics of group actions on rooted trees
R. I. Grigorchukab
a Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia
b Department of Mathematics, Texas A&M University, College Station, TX, USA
Abstract: This article combines the features of a survey and a research paper. It presents a review of some results obtained during the last decade in problems related to the dynamics of branch and self-similar groups on the boundary of a spherically homogeneous rooted tree and to the combinatorics and asymptotic properties of Schreier graphs associated with a group or with its action. Special emphasis is placed on the study of essentially free actions of self-similar groups, which are antipodes to branch actions. At the same time, the theme "free versus nonfree" runs through the paper. Sufficient conditions are obtained for the essential freeness of an action of a self-similar group on the boundary of a tree. Specific examples of such actions are given. Constructions of the associated dynamical system and the Schreier dynamical system generated by a Schreier graph are presented. For groups acting on trees, a trace on the associated $C^*$-algebra generated by a Koopman representation is introduced, and its role in the study of von Neumann factors, the spectral properties of groups, Schreier graphs, and elements of the associated $C^*$-algebra is demonstrated. The concepts of asymptotic expander and asymptotic Ramanujan graph are introduced, and examples of such graphs are given. Questions related to the notion of the cost of action and the notion of rank gradient are discussed.
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Proceedings of the Steklov Institute of Mathematics, 2011, 273, 64–175
Bibliographic databases:
UDC: 512+517.98+519.1
Received in May 2010
Citation: R. I. Grigorchuk, "Some topics in the dynamics of group actions on rooted trees", Modern problems of mathematics, Collected papers. In honor of the 75th anniversary of the Institute, Tr. Mat. Inst. Steklova, 273, MAIK Nauka/Interperiodica, Moscow, 2011, 72–191; Proc. Steklov Inst. Math., 273 (2011), 64–175
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This publication is cited in the following articles:
A. M. Vershik, "Totally nonfree actions and the infinite symmetric group", Mosc. Math. J., 12:1 (2012), 193–212
Dudko A., Medynets K., "On Characters of Inductive Limits of Symmetric Groups", J. Funct. Anal., 264:7 (2013), 1565–1598
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Bowen L., "Random Walks on Random Coset Spaces with Applications to Furstenberg Entropy", Invent. Math., 196:2 (2014), 485–510
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, ed. Morse R. NikolovaPopova D. Witherspoon S., Amer Mathematical Soc, 2014, 9–48
D'Angeli D., "Schreier Graphs of An Extended Version of the Binary Adding Machine", Electron. J. Comb., 21:4 (2014), P4.20
D'Angeli D., Rodaro E., "Groups and Semigroups Defined By Colorings of Synchronizing Automata", Int. J. Algebr. Comput., 24:6 (2014), 773–793
Grigorchuk R. Kravchenko R., "on the Lattice of Subgroups of the Lamplighter Group", Int. J. Algebr. Comput., 24:6 (2014), 837–877
Savchuk D., "Schreier Graphs of Actions of Thompson'S Group F on the Unit Interval and on the Cantor Set", Geod. Dedic., 175:1 (2015), 355–372
Malyshev A., Pak I., "Growth in Product Replacement Graphs of Grigorchuk Groups", J. Group Theory, 18:2 (2015), 209–234
D'Angeli D., Rodaro E., "a Geometric Approach To (Semi)-Groups Defined By Automata Via Dual Transducers", Geod. Dedic., 174:1 (2015), 375–400
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
Agnieszka Bier, Vitaliy Sushchansky, "Kaluzhnin's representations of Sylow $p$-subgroups of automorphism groups of $p$-adic rooted trees", Algebra Discrete Math., 19:1 (2015), 19–38
Bondarenko I., "Self-Similar Groups and the Zig-Zag and Replacement Products of Graphs", J. Algebra, 434 (2015), 1–11
Bowen L., Grigorchuk R., Kravchenko R., "Invariant Random Subgroups of Lamplighter Groups", Isr. J. Math., 207:2 (2015), 763–782
Bowen L., "Invariant Random Subgroups of the Free Group", Group. Geom. Dyn., 9:3 (2015), 891–916
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D'Angeli D., Rodaro E., "Freeness of automaton groups vs boundary dynamics", J. Algebra, 462 (2016), 115–136
Woryna A., "The classification of abelian groups generated by time-varying automata and by Mealy automata over the binary alphabet", Inf. Comput., 249 (2016), 18–27
Bou-Rabee Kh. Leemann P.-H. Nagnibeda T., "Weakly maximal subgroups in regular branch groups", J. Algebra, 455 (2016), 347–357
Creutz D., Peterson J., "Stabilizers of ergodic actions of lattices and commensurators", Trans. Am. Math. Soc., 369:6 (2017), 4119–4166
Dudko A. Grigorchuk R., "On spectra of Koopman, groupoid and quasi-regular representations", J. Mod. Dyn., 11 (2017), 99–123
Bowen L. Grigorchuk R. Kravchenko R., "Characteristic random subgroups of geometric groups and free abelian groups of infinite rank", Trans. Am. Math. Soc., 369:2 (2017), 755–781
Malyshev A., Pak I., "Lifts, Derandomization, and Diameters of Schreier Graphs of Mealy Automata", Combinatorica, 37:4 (2017), 733–765
Dudko A. Grigorchuk R., "On Irreducibility and Disjointness of Koopman and Quasi-Regular Representations of Weakly Branch Groups", Modern Theory of Dynamical Systems: a Tribute to Dmitry Victorovich Anosov, Contemporary Mathematics, 692, ed. Katok A. Pesin Y. Hertz F., Amer Mathematical Soc, 2017, 51–66
Kalogeropoulos N., "An Entropy For Groups of Intermediate Growth", Adv. Math. Phys., 2017, 2863614
Shaikh A., Bhate H., "Zeta Functions of Finite Schreier Graphs and Their Zig-Zag Products", J. Algebra. Appl., 16:8 (2017), 1750151
Creutz D., "Stabilizers of Actions of Lattices in Products of Groups", Ergod. Theory Dyn. Syst., 37:4 (2017), 1133–1186
Grigorchuk R. Lenz D. Nagnibeda T., "Schreier Graphs of Grigorchuk'S Group and a Subshift Associated to a Nonprimitive Substitution", Groups, Graphs and Random Walks, London Mathematical Society Lecture Note Series, 436, ed. CeccheriniSilberstein T. Salvatori M. SavaHuss E., Cambridge Univ Press, 2017, 250–299
Grigorchuk R. Musat M. Rordam M., "Just-Infinite $C^*$-Algebras", Comment. Math. Helv., 93:1 (2018), 157–201
Dudko A., Grigorchuk R., "On Diagonal Actions of Branch Groups and the Corresponding Characters", J. Funct. Anal., 274:11 (2018), 3033–3055
Salaff-Coste L., Zheng T., "Isoperimetric Profiles and Random Walks on Some Permutation Wreath Products", Rev. Mat. Iberoam., 34:2 (2018), 481–540
D'Angeli D., Rodaro E., "Fragile Words and Cayley Type Transducers", Int. J. Group Theory, 7:3, 3 (2018), 95–109
Grigorchuk R. Kravchenko R., "On the Rigidity of Rank Gradient in a Group of Intermediate Growth", Ukr. Math. J., 70:2 (2018), 182–196
Dudko A., "On Irreducibility of Koopman Representations Corresponding to Measure Contracting Actions", Group. Geom. Dyn., 12:4 (2018), 1417–1427
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