The braided Ptolemy Thompson group is asynchronously combable

The braided Ptolemy-Thompson group is asynchronously combable Louis Funar Christophe Kapoudjian Institut Fourier BP 74, UMR 5582 Laboratoire Emile Picard, UMR 5580 University of Grenoble I University of Toulouse III 38402 Saint-Martin-d'Heres cedex, France 31062 Toulouse cedex 4, France e-mail: e-mail: December 4, 2009 Abstract The braided Ptolemy-Thompson group T ? is an extension of the Thompson group T by the full braid group B∞ on infinitely many strands and both of them can be viewed as mapping class groups of certain infinite planar surfaces. The main result of this article is that T ? (and in particular T ) is asynchronously combable. The result is new already for the group T . The method of proof is inspired by Lee Mosher's proof of automaticity of mapping class groups. 2000 MSC Classification: 57 N 05, 20 F 38, 57 M 07, 20 F 34. Keywords: mapping class groups, infinite surface, Thompson group, braid. 1 Introduction 1.1 Statements and results The Thompson groups T and V were the first examples of finitely presented infinite simple groups. We refer to [9] for a survey concerning their classical properties. An algebraic relation between T and the braid groups has been discovered in an article due to P.

• d?


• thompson groups


• homeomorphisms modulo


• between ?


• infinite braid


• binary trees


• isotopy through