no. 1
Pentagon representations and complex projective structures on closed surfaces
Le Fils, Thomas1
1 Institut de Mathématiques de Jussieu Paris Rive Gauche (IMJ-PRG) 75005 Paris (France)
Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 423-445.

We define a class of representations of the fundamental group of a closed surface of genus 2 to PSL 2 (ℂ): the pentagon representations. We show that they are exactly the non-elementary PSL 2 (ℂ)-representations of surface groups that do not admit a Schottky decomposition, i.e. a pants decomposition such that the restriction of the representation to each pair of pants is an isomorphism onto a Schottky group. In doing so, we exhibit a gap in the proof of Gallo, Kapovich and Marden that every non-elementary representation of a surface group to PSL 2 (ℂ) is the holonomy of a projective structure, possibly with one branched point of order 2. We show that pentagon representations arise as such holonomies and repair their proof.

Nous définissons une classe de représentations du groupe fondamental d’une surface fermée de genre 2 dans PSL 2 (ℂ)  : les représentations pentagones. Nous montrons que ce sont exactement les représentations d’un groupe de surface dans PSL 2 (ℂ) qui n’admettent pas de décomposition de Schottky, i.e. de décomposition en pantalons telle que la restriction de la représentation à chaque pantalon est un isomorphisme sur un groupe de Schottky. Ce faisant, nous exhibons une lacune dans la preuve de Gallo, Kapovich et Marden du fait que toute représentation non-élémentaire d’un groupe de surface dans PSL 2 (ℂ) est l’holonomie d’une structure projective, avec éventuellement un point de branchement d’ordre 2. Nous montrons que les représentations pentagones sont de telles holonomies et réparons leur preuve.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3528
Classification: 57M50
Keywords: Projective structure, holonomy, surface groups representations.
Mot clés : Structure projective, holonomie, représentations de groupes de surfaces.

Le Fils, Thomas 1

1 Institut de Mathématiques de Jussieu Paris Rive Gauche (IMJ-PRG) 75005 Paris (France)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Le Fils, Thomas. Pentagon representations and complex projective structures on closed surfaces. Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 423-445. doi : 10.5802/aif.3528. https://aif.centre-mersenne.org/articles/10.5802/aif.3528/

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