# Groupes de Thompson projectifs de genre 0

Abstract : The Thompson projective group $T$ is the set of piecewise projective homeomorphisms of $\partial (\bf H)$ with rational breakpoints. For $\Gamma$ a subgroup of $PSL_2((\bf Z))$ we can consider the subgroup $T_(\Gamma)$ of $T$ of piecewise $\Gamma$ homeomorphisms, and we ask if the fundamental property of $T$ of being finitely generated is preserved. It depends on the genus of the associated surface. The main goal of our work is to prove that, when the genus is 0, $T_(\Gamma)$ is finitely presented (Peter Greenberg proved that when the genus is strictly positive, $T_(\Gamma)$ is not finitely generated). We start by proving that $T_(\Gamma)$ is conjugated to a group generated by two classical Thompson affine groups. Then we give a combinatorial description of $T_(\Gamma)$ with couples of infinite forests, and that permits us to find an infinite regular presentation of the group, and then a finite one.
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https://tel.archives-ouvertes.fr/tel-00007108
Contributor : Arlette Guttin-Lombard <>
Submitted on : Thursday, October 14, 2004 - 8:51:15 AM
Last modification on : Tuesday, May 11, 2021 - 11:36:03 AM
Long-term archiving on: : Monday, September 6, 2010 - 11:32:50 AM

### Identifiers

• HAL Id : tel-00007108, version 1

### Citation

Guillaume Laget. Groupes de Thompson projectifs de genre 0. Mathématiques [math]. Université Joseph-Fourier - Grenoble I, 2004. Français. ⟨tel-00007108⟩

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