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Mat. Zametki, 2002, Volume 71, Issue 3, Pages 334–347 (Mi mz350)  

This article is cited in 8 scientific papers (total in 8 papers)

On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line

L. A. Beklaryan

Central Economics and Mathematics Institute, RAS

Abstract: In [1] G. Margulis proved Ghys's conjecture stating the validity of the following analog of the Tits alternative: either the group $G\subseteq \operatorname {Homeo}(S^1)$ of homeomorphisms of the circle possesses a free subgroup with two generators or there is an invariant probabilistic measure on $S^1$. In the present paper, we prove the following strengthening of Margulis's statement: an invariant probabilistic measure for a group $G\subseteq \operatorname {Homeo}(S^1)$ exists if and only if the quotient group $G/H_G$ does not contain a free subgroup with two generators (here $H_G$ is some specific subgroup of $G$ defined in a canonical way). We also formulate and prove analogs of the Tits alternative for groups $G\subseteq \operatorname {Homeo}(\mathbb R)$ of homeomorphisms of the line.

DOI: https://doi.org/10.4213/mzm350

Full text: PDF file (229 kB)
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English version:
Mathematical Notes, 2002, 71:3, 305–315

Bibliographic databases:

UDC: 515.1
Received: 29.03.2001
Revised: 29.08.2001

Citation: L. A. Beklaryan, “On Analogs of the Tits Alternative for Groups of Homeomorphisms of the Circle and of the Line”, Mat. Zametki, 71:3 (2002), 334–347; Math. Notes, 71:3 (2002), 305–315

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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. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Navas A., “Solvable grounds of diffeomorphisms of the interval, circle and the real line”, Bull. Braz. Math. Soc. (N.S.), 35:1 (2004), 13–50  crossref  mathscinet  zmath  isi  scopus  scopus
    3. L. A. Beklaryan, “The structure of a group quasisymmetrically conjugate to a group of affine transformations of the real line”, Sb. Math., 196:10 (2005), 1403–1420  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Navas A., “Growth of groups and diffeomorphisms of the interval”, Geom. Funct. Anal., 18:3 (2008), 988–1028  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Bleak C., Kassabov M., Matucci F., “Structure Theorems for Groups of Homeomorphisms of the Circle”, Int. J. Algebr. Comput., 21:6 (2011), 1007–1036  crossref  mathscinet  zmath  isi  scopus  scopus
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    7. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    8. L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
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