#### Volume 12, issue 1 (2008)

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The braided Ptolemy–Thompson group is finitely presented

### Louis Funar and Christophe Kapoudjian

Geometry & Topology 12 (2008) 475–530
##### Abstract

Pursuing our investigations on the relations between Thompson groups and mapping class groups, we introduce the group ${T}^{♯}$ (and its companion ${T}^{\ast }$) which is an extension of the Ptolemy–Thompson group $T$ by the braid group ${B}_{\infty }$ 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 ${T}^{\ast }$ are in the same relation with respect to each other as the braid groups ${B}_{n+1}$ and ${B}_{n}$, for infinitely many strands $n$. We show that both groups embed as groups of homeomorphisms of the circle and their word problem is solvable.

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