Groups of line and circle homeomorphisms. Criteria for almost nilpotency. (English. Russian original) Zbl 1415.37055
Sb. Math. 210, No. 4, 495-507 (2019); translation from Mat. Sb. 210, No. 4, 27-40 (2019).
Summary: 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). A. Navas [Groups of circle diffeomorphisms. Chicago, IL: University of Chicago Press (2011; Zbl 1236.37002)]
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.
MSC:
| 37E10 | Dynamical systems involving maps of the circle |
| 37E05 | Dynamical systems involving maps of the interval |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
Keywords:
almost nilpotency; group of line homeomorphisms; group of circle homeomorphisms; free subsemigroupCitations:
Zbl 1236.37002References:
| [1] | L. A. Beklaryan 2012 Group specialties in the problem of the maximum principle for systems with deviating argument J. Dyn. Control Syst.18 3 419-432 · Zbl 1255.49038 · doi:10.1007/s10883-012-9151-6 |
| [2] | L. A. Beklaryan 2016 Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems Mat. Sb.207 8 47-72 · Zbl 1385.37052 · doi:10.4213/sm8596 |
| [3] | English transl. L. A. Beklaryan 2016 Sb. Math.207 8 1079-1099 · Zbl 1385.37052 · doi:10.1070/SM8596 |
| [4] | L. A. Beklaryan 2012 Residual subsets in the space of finitely generated groups of diffeomorphisms of the circle Mat. Zametki92 6 825-833 · doi:10.4213/mzm9151 |
| [5] | English transl. in L. A. Beklaryan 2013 Math. Notes93 1 29-35 · Zbl 1314.54025 · doi:10.1134/S0001434613010033 |
| [6] | L. A. Beklaryan Residual subsets of the space of finitely generated groups of \(C^{(1)}\)-homeomorphisms of a line or a circle Fundam. Prikl. Mat. · Zbl 1314.54025 |
| [7] | A. Navas 2011 Groups of circle diffeomorphisms Chicago Lectures in Math. Univ. of Chicago Press, Chicago, IL xviii+290 pp. · Zbl 1236.37002 · doi:10.7208/chicago/9780226569505.001.0001 |
| [8] | J. M. Rosenblatt 1974 Invariant measures and growth conditions Trans. Amer. Math. Soc.193 33-53 · Zbl 0246.28017 · doi:10.1090/S0002-9947-1974-0342955-9 |
| [9] | M. Gromov 1981 Groups of polynomial growth and expanding maps Inst. Hautes Études Sci. Publ. Math.53 1 53-78 · Zbl 0474.20018 · doi:10.1007/BF02698687 |
| [10] | M. I. Kargapolov and Ju. I. Merzljakov 1977 Fundamentals of the theory of groups Nauka, Moscow 2nd ed., 240 pp. |
| [11] | English transl. M. I. Kargapolov and Ju. I. Merzljakov 1979 Grad. Texts in Math. 62 Springer-Verlag, New York-Berlin xvii+203 pp. · Zbl 0549.20001 |
| [12] | A. I. Malcev 1976 Selected works 1 Classical algebra Nauka, Moscow 484 pp. · Zbl 0422.01024 |
| [13] | É. Ghys 2001 Groups acting on the circle Enseign. Math. (2)47 3-4 329-407 · Zbl 1044.37033 |
| [14] | B. Farb and J. Franks 2003 Groups of homeomorphisms of one-manifolds. III. Nilpotent subgroups Ergodic Theory Dynam. Systems23 5 1467-1484 · Zbl 1037.37020 · doi:10.1017/S0143385702001712 |
| [15] | B. Deroin, V. Kleptsyn and A. Navas 2007 Sur la dynamique unidimensionnelle en régularité intermédiaire Acta Math.199 2 199-262 · Zbl 1139.37025 · doi:10.1007/s11511-007-0020-1 |
| [16] | J. F. Plante and W. P. Thurston 1976 Polynomial growth in holonomy groups of foliations Comment. Math. Helv.51 567-584 · Zbl 0348.57009 · doi:10.1007/BF02568174 |
| [17] | R. Baer 1956 Noethersche Gruppen Math. Z.66 269-288 · Zbl 0072.25802 · doi:10.1007/BF01186613 |
| [18] | L. A. Beklaryan 1996 On the classification of groups of orientation-preserving homeomorphisms of \(\mathbb R\). II. Projectively-invariant measures Mat. Sb.187 4 3-28 · Zbl 0870.54041 · doi:10.4213/sm121 |
| [19] | English transl. L. A. Beklaryan 1996 Sb. Math.187 4 469-494 · Zbl 0870.54041 · doi:10.1070/SM1996v187n04ABEH000121 |
| [20] | I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai 1980 Ergodic theory Nauka, Moscow 384 pp. · Zbl 0508.28008 |
| [21] | English transl. I. P. Cornfeld, S. V. Fomin and Ya. G. Sinai 1982 Grundlehren Math. Wiss. 245 Springer-Verlag, New York x+486 pp. · Zbl 0493.28007 · doi:10.1007/978-1-4615-6927-5 |
| [22] | L. V. Ahlfors 1981 Moebius transformations in several dimensions Ordway Professorship Lectures Math. Univ. of Minnesota, School of Mathematics, Minneapolis, MN ii+150 pp. · Zbl 0517.30001 |
| [23] | L. A. Beklaryan 2004 Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants Uspekhi Mat. Nauk59 4(358) 3-68 · Zbl 1073.54018 · doi:10.4213/rm758 |
| [24] | English transl. in L. A. Beklaryan 2004 Russian Math. Surveys59 4 599-660 · Zbl 1073.54018 · doi:10.1070/RM2004v059n04ABEH000758 |
| [25] | L. A. Beklaryan 1993 Invariant and projectively invariant measures for groups of orientation-preserving homeomorphisms of \(\mathbb{R} \) Dokl. Ross. Akad. Nauk332 6 679-681 · Zbl 0837.57027 |
| [26] | English transl. in L. A. Beklaryan 1994 Russian Acad. Sci. Dokl. Math.48 2 387-390 |
| [27] | L. A. Beklaryan 2002 On analogs of the Tits alternative for groups of homeomorphisms of the circle and of the line Mat. Zametki71 3 334-347 · Zbl 1024.22012 · doi:10.4213/mzm350 |
| [28] | English transl. in L. A. Beklaryan 2002 Math. Notes71 3 305-315 · Zbl 1024.22012 · doi:10.1023/A:1014890606194 |
| [29] | L. A. Beklaryan 2015 Groups of line and circle homeomorphisms. Metric invariants and questions of classification Uspekhi Mat. Nauk70 2(422) 3-54 · Zbl 1352.37110 · doi:10.4213/rm9654 |
| [30] | English transl. in L. A. Beklaryan 2015 Russian Math. Surveys70 2 203-248 · Zbl 1352.37110 · doi:10.1070/RM2015v070n02ABEH004946 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
