Algebraic & Geometric Topology

On periodic groups of homeomorphisms of the $2$–dimensional sphere

Jonathan Conejeros

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Abstract

We prove that every finitely generated group of homeomorphisms of the 2 –dimensional sphere all of whose elements have a finite order which is a power of 2 and is such that there exists a uniform bound for the orders of the group elements is finite. We prove a similar result for groups of area-preserving homeomorphisms without the hypothesis that the orders of group elements are powers of 2 provided there is an element of even order.

Article information

Source
Algebr. Geom. Topol., Volume 18, Number 7 (2018), 4093-4107.

Dates
Received: 1 December 2017
Revised: 7 June 2018
Accepted: 14 July 2018
First available in Project Euclid: 18 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1545102065

Digital Object Identifier
doi:10.2140/agt.2018.18.4093

Mathematical Reviews number (MathSciNet)
MR3892240

Zentralblatt MATH identifier
07006386

Subjects
Primary: 20F50: Periodic groups; locally finite groups 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 37E30: Homeomorphisms and diffeomorphisms of planes and surfaces 37E45: Rotation numbers and vectors 57S25: Groups acting on specific manifolds

Keywords
Burnside problem surface homeomorphisms $2$–sphere

Citation

Conejeros, Jonathan. On periodic groups of homeomorphisms of the $2$–dimensional sphere. Algebr. Geom. Topol. 18 (2018), no. 7, 4093--4107. doi:10.2140/agt.2018.18.4093. https://projecteuclid.org/euclid.agt/1545102065


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References

  • W Burnside, On an unsettled question in the theory of discontinuous groups, Quart. J. Math. 33 (1902) 230–238
  • A Constantin, B Kolev, The theorem of Kerékjártó on periodic homeomorphisms of the disc and the sphere, Enseign. Math. 40 (1994) 193–204
  • E Ghys, Groups acting on the circle, Enseign. Math. 47 (2001) 329–407
  • E S Golod, On nil-algebras and finitely approximable $p$–groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 273–276 In Russian; translated in “Fourteen papers on logic, algebra, complex variables and topology”, Amer. Math. Soc. Transl. Ser. (2) 48, Amer. Math. Soc., Providence RI (1965) 103–106
  • N Guelman, I Liousse, Burnside problem for measure preserving groups and for $2$–groups of toral homeomorphisms, Geom. Dedicata 168 (2014) 387–396
  • N Guelman, I Liousse, Burnside problem for groups of homeomorphisms of compact surfaces, Bull. Braz. Math. Soc. 48 (2017) 389–397
  • S Hurtado, The Burnside problem for $\mathrm{Diff}_{\mathrm{Vol}}(\mathbb{S}^2)$, preprint (2016)
  • S V Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994) 1–308
  • B von Kerékjártó, Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann. 80 (1919) 36–38
  • P Le Calvez, Rotation numbers in the infinite annulus, Proc. Amer. Math. Soc. 129 (2001) 3221–3230
  • F Le Roux, L'ensemble de rotation autour d'un point fixe, Astérisque 350, Soc. Math. France, Paris (2013)
  • I G Lysënok, Infinite Burnside groups of even period, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996) 3–224 In Russian; translated in Izv. Math. 60 (1996) 453–654
  • A Navas, Groups of circle diffeomorphisms, Univ. Chicago Press (2011)
  • A Y Ol'shanskiĭ, The Novikov–Adyan theorem, Mat. Sb. 118 (1982) 203–235 In Russian; translated in Math. USSR-Sb. 46 (1983) 203–236