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The braided Ptolemy–Thompson group is finitely presented

Louis Funar and Christophe Kapoudjian

Geometry & Topology 12 (2008) 475–530

Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group T (and its companion T) which is an extension of the Ptolemy–Thompson group T by the braid group B on infinitely many strands. We prove that T is a finitely presented group by constructing a complex on which it acts cocompactly with finitely presented stabilizers, and derive from it an explicit presentation. The groups T and T are in the same relation with respect to each other as the braid groups Bn+1 and Bn, for infinitely many strands n. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable.

braid groups, mapping class groups, infinite surface, Thompson group
Mathematical Subject Classification 2000
Primary: 20F36, 57M07
Secondary: 20F38, 20F05, 57N05
Received: 26 June 2007
Accepted: 21 November 2007
Published: 12 March 2008
Proposed: Shigeyuki Morita
Seconded: Joan Birman, Jean-Pierre Otal
Louis Funar
Institut Fourier BP 74, UMR 5582
University of Grenoble I
38402 Saint-Martin-d’Hères cedex
Christophe Kapoudjian
Laboratoire Emile Picard, UMR 5580
University of Toulouse III
31062 Toulouse cedex 4