Remote Access Transactions of the American Mathematical Society Series B
Gold Open Access

Transactions of the American Mathematical Society Series B

ISSN 2330-0000

   
 
 

 

Uniform simplicity of groups with proximal action


Authors: Światosław R. Gal and Jakub Gismatullin; with an appendix by Nir Lazarovich
Journal: Trans. Amer. Math. Soc. Ser. B 4 (2017), 110-130
MSC (2010): Primary 20E08, 20E32; Secondary 20F65, 22E40
DOI: https://doi.org/10.1090/btran/18
Published electronically: September 6, 2017
Full-text PDF Open Access
View in AMS MathViewer New

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that groups acting boundedly and order-primitively on linear orders or acting extremely proximally on a Cantor set (the class including various Higman-Thomson groups; Neretin groups of almost automorphisms of regular trees, also called groups of spheromorphisms; the groups of quasi-isometries and almost-isometries of regular trees) are uniformly simple. Explicit bounds are provided.


References [Enhancements On Off] (What's this?)

  • [1] R. D. Anderson, On homeomorphisms as products of conjugates of a given homeomorphism and its inverse, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961) Prentice-Hall, Englewood Cliffs, N.J., 1962, pp. 231-234. MR 0139684
  • [2] Valery Bardakov, Vladimir Tolstykh, and Vladimir Vershinin, Generating groups by conjugation-invariant sets, J. Algebra Appl. 11 (2012), no. 4, 1250071, 16. MR 2959420, https://doi.org/10.1142/S0219498812500715
  • [3] Robert Bieri and Ralph Strebel, On groups of PL-homeomorphisms of the real line, Mathematical Surveys and Monographs, vol. 215, American Mathematical Society, Providence, RI, 2016. MR 3560537
  • [4] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486
  • [5] Kenneth S. Brown, Finiteness properties of groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 45-75. MR 885095, https://doi.org/10.1016/0022-4049(87)90015-6
  • [6] Dmitri Burago and Sergei Ivanov, A remark on the group of PL-homeomorphisms in dimension one, Geometric and probabilistic structures in dynamics, Contemp. Math., vol. 469, Amer. Math. Soc., Providence, RI, 2008, pp. 141-148. MR 2478468, https://doi.org/10.1090/conm/469/09164
  • [7] Dmitri Burago, Sergei Ivanov, and Leonid Polterovich, Conjugation-invariant norms on groups of geometric origin, Groups of diffeomorphisms, Adv. Stud. Pure Math., vol. 52, Math. Soc. Japan, Tokyo, 2008, pp. 221-250. MR 2509711
  • [8] Marc Burger and Shahar Mozes, Groups acting on trees: from local to global structure, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 113-150 (2001). MR 1839488
  • [9] Danny Calegari, Stable commutator length in subgroups of $ {\rm PL}^+(I)$, Pacific J. Math. 232 (2007), no. 2, 257-262. MR 2366352, https://doi.org/10.2140/pjm.2007.232.257
  • [10] Pierre-Emmanuel Caprace, Automorphism groups of right-angled buildings: simplicity and local splittings, Fund. Math. 224 (2014), no. 1, 17-51. MR 3164745, https://doi.org/10.4064/fm224-1-2
  • [11] Pierre-Emmanuel Caprace and Koji Fujiwara, Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups, Geom. Funct. Anal. 19 (2010), no. 5, 1296-1319. MR 2585575, https://doi.org/10.1007/s00039-009-0042-2
  • [12] C. G. Chehata, An algebraically simple ordered group, Proc. London Math. Soc. (3) 2 (1952), 183-197. MR 0047031, https://doi.org/10.1112/plms/s3-2.1.183
  • [13] Manfred Droste and R. M. Shortt, Commutators in groups of order-preserving permutations, Glasgow Math. J. 33 (1991), no. 1, 55-59. MR 1089954, https://doi.org/10.1017/S001708950000803X
  • [14] Gábor Elek and Endre Szabó, Hyperlinearity, essentially free actions and $ L^2$-invariants. The sofic property, Math. Ann. 332 (2005), no. 2, 421-441. MR 2178069, https://doi.org/10.1007/s00208-005-0640-8
  • [15] Erich W. Ellers, Nikolai Gordeev, and Marcel Herzog, Covering numbers for Chevalley groups, Israel J. Math. 111 (1999), 339-372. MR 1710745, https://doi.org/10.1007/BF02810691
  • [16] Światosław R. Gal and Jarek Kedra, On bi-invariant word metrics, J. Topol. Anal. 3 (2011), no. 2, 161-175. MR 2819193, https://doi.org/10.1142/S1793525311000556
  • [17] L. Garncarek and N. Lazarovich, The Neretin groups, arXiv:1502.00991v2, 2015.
  • [18] Thierry Giordano, Ian F. Putnam, and Christian F. Skau, Full groups of Cantor minimal systems, Israel J. Math. 111 (1999), 285-320. MR 1710743, https://doi.org/10.1007/BF02810689
  • [19] Jakub Gismatullin, Boundedly simple groups of automorphisms of trees, J. Algebra 392 (2013), 226-243. MR 3085032, https://doi.org/10.1016/j.jalgebra.2013.06.023
  • [20] Shmuel Glasner, Topological dynamics and group theory, Trans. Amer. Math. Soc. 187 (1974), 327-334. MR 0336723, https://doi.org/10.2307/1997056
  • [21] Shmuel Glasner, Proximal flows, Lecture Notes in Mathematics, Vol. 517, Springer-Verlag, Berlin-New York, 1976. MR 0474243
  • [22] Nikolai Gordeev and Jan Saxl, Products of conjugacy classes in Chevalley groups. I. Extended covering numbers, Israel J. Math. 130 (2002), 207-248. MR 1919378, https://doi.org/10.1007/BF02764077
  • [23] N. L. Gordeev, Products of conjugacy classes in algebraic groups. I, II, J. Algebra 173 (1995), no. 3, 715-744, 745-779. MR 1327877, https://doi.org/10.1006/jabr.1995.1111
  • [24] Charles Holland, Transitive lattice-ordered permutation groups, Math. Z. 87 (1965), 420-433. MR 0178052, https://doi.org/10.1007/BF01111722
  • [25] Christophe Kapoudjian, Simplicity of Neretin's group of spheromorphisms, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 4, 1225-1240. MR 1703086
  • [26] Stephen H. McCleary, Lattice-ordered permutation groups: the structure theory, Ordered groups and infinite permutation groups, Math. Appl., vol. 354, Kluwer Acad. Publ., Dordrecht, 1996, pp. 29-62. MR 1486196
  • [27] Yu. A. Neretin, Combinatorial analogues of the group of diffeomorphisms of the circle, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 5, 1072-1085; English transl., Russian Acad. Sci. Izv. Math. 41 (1993), no. 2, 337-349. MR 1209033, https://doi.org/10.1070/IM1993v041n02ABEH002264
  • [28] Jacques Tits, Sur le groupe des automorphismes d'un arbre, Essays on topology and related topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 188-211. MR 0299534

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society, Series B with MSC (2010): 20E08, 20E32, 20F65, 22E40

Retrieve articles in all journals with MSC (2010): 20E08, 20E32, 20F65, 22E40


Additional Information

Światosław R. Gal
Affiliation: Instytut Matematyczny, Uniwersytetu Wrocławskiego, pl. Grunwaldzki \nicefrac24, 50-384 Wrocław, Poland – and – Weizmann Institute of Science, Rehovot 76100, Israel
Email: sgal@math.uni.wroc.pl

Jakub Gismatullin
Affiliation: Instytut Matematyczny, Uniwersytetu Wrocławskiego, pl. Grunwaldzki \nicefrac24, 50-384 Wrocław, Poland – and – Instytut Matematyczny, Polskiej Akademii Nauk, ul. Śniadeckich 8, 00-656 Warszawa, Poland
Email: gismat@math.uni.wroc.pl

Nir Lazarovich
Affiliation: Departement Mathematik, Eidgenössische Technische Hochschule Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Email: nir.lazarovich@math.ethz.ch

DOI: https://doi.org/10.1090/btran/18
Keywords: Boundedly simple groups, trees, automorphism groups, spheromorphisms, almost automorphisms, Higman-Thomson groups, Neretin group
Received by editor(s): June 14, 2016
Received by editor(s) in revised form: February 27, 2017
Published electronically: September 6, 2017
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291111. The first author partially supported by Polish National Science Center (NCN) grant 2012/06/A/ST1/00259 and the European Research Council grant No. 306706.
The second author is partially supported by the NCN grants 2014/13/D/ST1/03491, 2012/07/B/ST1/03513.
Article copyright: © Copyright 2017 by the author under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)

American Mathematical Society