Geodesic
Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems
Sbornik. Mathematics, Tome 207 (2016) no. 8, pp. 1079-1099 Cet article a éte moissonné depuis la source Math-Net.Ru

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Almost nilpotency criteria and structure theorems are presented for the class of finitely generated groups of line and circle diffeomorphisms with mutually transversal elements. Key ingredients in the proof of the structure theorems are the existence/absence of an invariant measure, the (previously established) criterion for the existence of an invariant measure and restatements of this criterion in terms of various (topological, algebraic, combinatorial) characteristics of the group. The question of whether certain features of these characteristics or the existence of an invariant measure are typical for groups of line and circle diffeomorphisms is discussed. Bibliography: 34 titles.
Keywords: groups of diffeomorphisms, almost nilpotency criterion, invariant measure. 
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L. A. Beklaryan. Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems. Sbornik. Mathematics, Tome 207 (2016) no. 8, pp. 1079-1099. http://geodesic.mathdoc.fr/item/SM_2016_207_8_a2/

[1] M. M. Krylov, M. M. Bogolyubov, “Zagalna teoriya miri ta ii zastosuvaniya do vivcheniya dinamichnikh sistem neliniinikh mekhanitsi”, Zbirnik prats z neliniinoi mekhaniki, Zapiski kafedri matematichnoi fiziki Institutu budivelnoi mekhaniki AN URSR, 3, 1937, 55–112

[2] M. Day, “Amenable semigroups”, Illinois J. Math., 1:4 (1957), 509–544 | MR | Zbl

[3] J. F. Plante, “Foliations with measure preserving holonomy”, Ann. of Math. (2), 102:2 (1975), 327–361 | DOI | MR | Zbl

[4] M. Gromov, “Groups of polinomial growth and expending maps”, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 53–78 | DOI | MR | Zbl

[5] A. Navas, Groups of circle diffeomorphisms, Chicago Lectures in Math., Univ. of Chicago Press, Chicago, IL, 2011, xviii+290 pp. | DOI | MR | Zbl

[6] B. Deroin, V. Kleptsyn, A. Navas, “Sur la dynamique unidimensionnelle en régularité intermédiaire”, Acta Math., 199:2 (2007), 199–262 | DOI | MR | Zbl

[7] R. I. Grigorchuk, A. Maki, “A group of intermediate growth acting by homomorphisms on the real line”, Math. Notes, 53:2 (1993), 146–157 | DOI | MR | Zbl

[8] L. A. Beklaryan, “Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants”, Russian Math. Surveys, 59:4 (2004), 599–660 | DOI | DOI | MR | Zbl

[9] L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248 | DOI | DOI | MR | Zbl

[10] P. de la Harpe, R. I. Grigorchuk, T. Ceccherini-Silberstein, “Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces”, Proc. Steklov Inst. Math., 224 (1999), 57–97 | MR | Zbl

[11] S. I. Adian, “New estimates of odd exponents of infinite Burnside groups”, Proc. Steklov Inst. Math., 289 (2015), 33–71 | DOI | DOI | Zbl

[12] L. A. Beklaryan, “On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. II. Projectively-invariant measures”, Sb. Math., 187:4 (1996), 469–494 | DOI | DOI | MR | Zbl

[13] J. F. Plante, “Solvable groups acting on the line”, Trans. Amer. Math. Soc., 278:1 (1983), 401–414 | DOI | MR | Zbl

[14] E. Salhi, “Sur les ensembles minimaux locaux”, C. R. Acad. Sci. Paris Sér. I Math., 295:12 (1982), 691–694 | MR | Zbl

[15] E. Salhi, “Sur un théorème de structure des feuilletages de codimension 1”, C. R. Acad. Sci. Paris Sér. I Math., 300:18 (1985), 635–638 | MR | Zbl

[16] E. Salhi, “Niveau de feuilles”, C. R. Acad. Sci. Paris Sér. I Math., 301:5 (1985), 219–222 | MR | Zbl

[17] L. A. Beklaryan, “On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. III. $\omega$-projectively invariant measures”, Sb. Math., 190:4 (1999), 521–538 | DOI | DOI | MR | Zbl

[18] I. P. Cornfeld, S. V. Fomin, Ya. G. Sinai, Ergodic theory, Grundlehren Math. Wiss., 245, Springer-Verlag, New York, 1982, x+486 pp. | DOI | MR | MR | Zbl | Zbl

[19] L. V. Ahlfors, Moebius transformations in several dimensions, Ordway Professorship Lectures Math., Univ. of Minnesota, School of Mathematics, Minneapolis, MI, 1981, ii+150 pp. | MR | MR | Zbl | Zbl

[20] L. A. Beklaryan, “On a criterion for the topological conjugacy of a quasisymmetric group to a group of affine transformations of $\mathbb R$”, Sb. Math., 191:6 (2000), 809–819 | DOI | DOI | MR | Zbl

[21] L. A. Beklaryan, “On analogs of the Tits alternative for groups of homeomorphisms of the circle and of the line”, Math. Notes, 71:3 (2002), 305–315 | DOI | DOI | MR | Zbl

[22] L. A. Beklaryan, “Invariant and projectively invariant measures for groups of orientation-preserving homeomorphisms of $\mathbb{R}$”, Russian Acad. Sci. Dokl. Math., 48:2 (1994), 387–390 | MR | Zbl

[23] L. A. Beklaryan, “A criterion connected with the structure of the fixed-point set for the existence of a projectively invariant measure for groups of orientation-preserving homeomorphisms of $\mathbb R$”, Russian Math. Surveys, 51:3 (1996), 539–540 | DOI | DOI | MR | Zbl

[24] V. V. Solodov, “Homeomorphisms of the circle and a foliation”, Math. USSR-Izv., 24:3 (1985), 553–566 | DOI | MR | Zbl

[25] G. Margulis, “Free subgroups of the homeomorphism group of the circle”, C. R. Acad. Sci. Paris Sér. I Math., 331:9 (2000), 669–674 | DOI | MR | Zbl

[26] L. A. Beklaryan, “Criteria for the existence of an invariant measure for groups of homeomorphisms of the line”, Math. Notes, 95:3 (2014), 304–307 | DOI | DOI | MR | Zbl

[27] L. A. Beklaryan, “Groups of homeomorphisms of the line. Criteria for the existence of invariant and projectively invariant measures in terms of the commutator subgroup”, Sb. Math., 205:12 (2014), 1741–1760 | DOI | DOI | MR | Zbl

[28] L. A. Beklaryan, “On the classification of groups of orientation-preserving homeomorphisms of $\mathbb R$. I. Invariant measures”, Sb. Math., 187:3 (1996), 335–364 | DOI | DOI | MR | Zbl

[29] D. V. Anosov, “On the contribution of N. N. Bogolyubov to the theory of dynamical systems”, Russian Math. Surveys, 49:5 (1994), 1–18 | DOI | MR | Zbl

[30] A. G. Kurosh, Lectures in general algebra, Internat. Ser. Monogr. Pure Appl. Math., 70, Pergamon Press, Oxford–Edinburgh–New York, 1965, x+364 pp. | MR | MR | Zbl | Zbl

[31] L. A. Beklaryan, “Residual subsets in the space of finitely generated groups of diffeomorphisms of the circle”, Math. Notes, 93:1 (2013), 29–35 | DOI | DOI | MR | Zbl

[32] J. M. Rosenblatt, “Invariant measures and growth conditions”, Trans. Amer. Math. Soc., 193 (1974), 33–53 | DOI | MR | Zbl

[33] L. A. Beklaryan, “Group specialities in the problem of the maximum principle for systems with deviating argument”, J. Dyn. Control Syst., 18:3 (2012), 419–432 | DOI | MR | Zbl

[34] V. Arnol'd, Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Mir, M., 1980, 324 pp. | MR | MR | Zbl | Zbl