Burnside problem for measure preserving groups and

Authors: Nancy Guelman Isabelle Liousse

Publish Date: 2013/02/15

Volume: 168, Issue: 1, Pages: 387-396

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Abstract

A group G is said to be periodic if for every gin G there exists a positive integer n with gn=mathrmId We prove that a finitely generated periodic group of homeomorphisms on the 2torus that preserves a probability measure mu is finite Moreover if the group consists of homeomorphisms isotopic to the identity then it is abelian and acts freely on mathbbT 2 In the Appendix we show that every finitely generated 2group of toral homeomorphisms is finiteWe are grateful to Andrés Navas for proposing to us this subject and for fruitful discussions We thank Christian Bonatti for suggesting us to add Proposition 3 We thank the referee for giving relevant and helpful comments suggestions and correctionsIn this section we prove that a finitely generated periodic group of isotopic to identity toral homeomorphisms whose generators have order 2 is finite and isomorphic to 1 mathbbZ /2mathbbZ or mathbbZ /2mathbbZ oplus mathbbZ /2mathbbZ This is a consequence of the fact that every element of order 2 in G 0 a periodic group of isotopic to identity toral homeomorphisms belongs to the center of G 0 It will be proven in Proposition 7Let G be a group generated by s elements g 1 g s of finite order an element of G can be written g= g 1p 1g sp s C where Cin GG and p ige 0 is bounded by the order of g i So the index of GG in G is finite moreover it is bounded by the product of the orders of g 1 g s

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