Geodesic
Groups of line and circle homeomorphisms. Criteria for almost nilpotency
Sbornik. Mathematics, Tome 210 (2019) no. 4, pp. 495-507 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For finitely-generated groups of line and circle homeomorphisms a criterion for their being almost nilpotent is established in terms of free two-generator subsemigroups and the condition of maximality. Previously the author found a criterion for almost nilpotency stated in terms of free two-generator subsemigroups for finitely generated groups of line and circle homeomorphisms that are $C^{(1)}$-smooth and mutually transversal. In addition, for groups of diffeomorphisms, structure theorems were established and a number of characteristics of such groups were proved to be typical. It was also shown that, in the space of finitely generated groups of $C^{(1)}$-diffeomorphisms with a prescribed number of generators, the set of groups with mutually transversal elements contains a countable intersection of open dense subsets (is residual). Navas has also obtained a criterion for the almost nilpotency of groups of $C^{(1+\alpha)}$-diffeomorphisms of an interval, where $\alpha>0$, in terms of free subsemigroups on two generators. Bibliography: 21 titles.
Keywords: almost nilpotency, group of line or circle homeomorphisms, free subsemigroup. 
@article{SM_2019_210_4_a1,
     author = {L. A. Beklaryan},
     title = {Groups of line and circle homeomorphisms. {Criteria} for almost nilpotency},
     journal = {Sbornik. Mathematics},
     pages = {495--507},
     year = {2019},
     volume = {210},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_4_a1/}
}
TY  - JOUR
AU  - L. A. Beklaryan
TI  - Groups of line and circle homeomorphisms. Criteria for almost nilpotency
JO  - Sbornik. Mathematics
PY  - 2019
SP  - 495
EP  - 507
VL  - 210
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2019_210_4_a1/
LA  - en
ID  - SM_2019_210_4_a1
ER  - 
%0 Journal Article
%A L. A. Beklaryan
%T Groups of line and circle homeomorphisms. Criteria for almost nilpotency
%J Sbornik. Mathematics
%D 2019
%P 495-507
%V 210
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2019_210_4_a1/
%G en
%F SM_2019_210_4_a1
L. A. Beklaryan. Groups of line and circle homeomorphisms. Criteria for almost nilpotency. Sbornik. Mathematics, Tome 210 (2019) no. 4, pp. 495-507. http://geodesic.mathdoc.fr/item/SM_2019_210_4_a1/

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

[2] L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099 | DOI | DOI | MR | Zbl

[3] 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

[4] L. A. Beklaryan, “O massivnykh podmnozhestvakh v prostranstve konechno-porozhdennykh grupp diffeomorfizmov pryamoi i okruzhnosti gladkosti $C^{(1)}$”, Fundament. i prikl. matem. (to appear)

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

[7] M. Gromov, “Groups of polynomial growth and expanding maps”, Inst. Hautes Études Sci. Publ. Math., 53:1 (1981), 53–78 | DOI | MR | Zbl

[8] M. I. Kargapolov, Ju. I. Merzljakov, Fundamentals of the theory of groups, Grad. Texts in Math., 62, Springer-Verlag, New York–Berlin, 1979, xvii+203 pp. | MR | MR | Zbl | Zbl

[9] A. I. Maltsev, Izbrannye trudy, v. 1, Klassicheskaya algebra, Nauka, M., 1976, 484 pp. | MR | Zbl

[10] É. Ghys, “Groups acting on the circle”, Enseign. Math. (2), 47:3-4 (2001), 329–407 | MR | Zbl

[11] B. Farb, J. Franks, “Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups”, Ergodic Theory Dynam. Systems, 23:5 (2003), 1467–1484 | DOI | MR | Zbl

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

[13] J. F. Plante, W. P. Thurston, “Polynomial growth in holonomy groups of foliations”, Comment. Math. Helv., 51 (1976), 567–584 | DOI | MR | Zbl

[14] R. Baer, “Noethersche Gruppen”, Math. Z., 66 (1956), 269–288 | DOI | MR | Zbl

[15] 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

[16] 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

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

[18] 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

[19] 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

[20] 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

[21] 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