The braided Ptolemy–Thompson group is finitely presented

Funar, Louis¹ ; Kapoudjian, Christophe²

¹ Institut Fourier BP 74, UMR 5582, University of Grenoble I, 38402 Saint-Martin-d’Hères cedex, France
² Laboratoire Emile Picard, UMR 5580, University of Toulouse III, 31062 Toulouse cedex 4, France

Geometry & topology, Tome 12 (2008) no. 1, pp. 475-530.

Résumé

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.

Détail

DOI : 10.2140/gt.2008.12.475

Keywords: braid groups, mapping class groups, infinite surface, Thompson group

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