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Short proofs of theorems of Malyutin and Margulis. (English) Zbl 1377.54043

This elegant paper uses ingenious ideas from topological dynamics to provide a particularly short journey to a theorem of A. V. Malyutin [St. Petersbg. Math. J. 19, No. 2, 279–296 (2008; Zbl 1209.37009); translation from Algebra Anal. 19, No. 2, 156–182 (2007)] and thence to a proof of the Ghys-Margulis alternative [G. Margulis, C. R. Acad. Sci., Paris, Sér. I, Math. 331, No. 9, 669–674 (2000; Zbl 0983.37029)] and [É. Ghys, Enseign. Math. (2) 47, No. 3–4, 329–407 (2001; Zbl 1044.37033)], which says that if \(G\) is a countable group of homeomorphisms of the circle acting minimally, then either \(G\) is isomorphic to a subgroup of \(\mathbb{T}\times\mathbb{Z}/2\mathbb{Z}\) or \(G\) contains a free group on two generators.

MSC:

54H20 Topological dynamics (MSC2010)
37B05 Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.)
20B07 General theory for infinite permutation groups
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References:

[1] Auslander, Joseph, Regular minimal sets. I, Trans. Amer. Math. Soc., 123, 469-479 (1966) · Zbl 0139.40901
[2] Beklaryan, L. A., Groups of homeomorphisms of the line and the circle. Topological characteristics and metric invariants, Uspekhi Mat. Nauk. Russian Math. Surveys, 59 59, 4, 599-660 (2004) · Zbl 1073.54018
[3] Ellis, Robert, Lectures on topological dynamics, xv+211 pp. (1969), W. A. Benjamin, Inc., New York · Zbl 0193.51502
[4] Ghys, \'Etienne, Groups acting on the circle, Enseign. Math. (2), 47, 3-4, 329-407 (2001) · Zbl 1044.37033
[5] Glasner, Shmuel, Proximal flows, Lecture Notes in Mathematics, Vol. 517, viii+153 pp. (1976), Springer-Verlag, Berlin-New York · Zbl 0322.54017
[6] Malyutin, A. V., Classification of group actions on the line and the circle, Algebra i Analiz. St. Petersburg Math. J., 19 19, 2, 279-296 (2008) · Zbl 1209.37009
[7] Margulis, Gregory, Free subgroups of the homeomorphism group of the circle, C. R. Acad. Sci. Paris S\'er. I Math., 331, 9, 669-674 (2000) · Zbl 0983.37029
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