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

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

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English version:
Mathematical Notes, 2002, 71:3, 305–315

Bibliographic databases:

UDC: 515.1
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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz350
• https://doi.org/10.4213/mzm350
• http://mi.mathnet.ru/eng/mz/v71/i3/p334

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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
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
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
4. Navas A., “Growth of groups and diffeomorphisms of the interval”, Geom. Funct. Anal., 18:3 (2008), 988–1028
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
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
7. L. A. Beklaryan, “Groups of line and circle homeomorphisms. Metric invariants and questions of classification”, Russian Math. Surveys, 70:2 (2015), 203–248
8. L. A. Beklaryan, “Groups of line and circle diffeomorphisms. Criteria for almost nilpotency and structure theorems”, Sb. Math., 207:8 (2016), 1079–1099
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