Pentagon representations and complex projective structures on closed surfaces

Abstract

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.Résumé. -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.

Holonomy of complex projective structures on surfaces with prescribed branch data

“…In the case where $$\rho$$ is a pentagon representation, there exists a branched projective structure on $$\Sigma _2$$ with a single conical point of total angle $$2\pi (1 + 1)$$ and holonomy $$\rho$$, see [26, Theorem 1.6]. Moreover, we can assume that there exists a curve based at the conical point that develops injectively, see the red curve in [26, figure 8], whose exponential develops injectively. Thus, we have $$\rho \in \mathrm{Hol}(\mathcal {P}(2d +1))$$ for every $$d\geqslant 0$$ by Proposition 5.7 and $$\rho \in \mathrm{Hol}(\mathcal {P}(n_1, \ldots , n_k))$$ if $$\sum _i n_i$$ is odd by Corollary 5.9.…”

“…Indeed, the techniques of construction of projective structures with given holonomy of Gallo, Kapovich, and Marden allow them to prove directly this theorem when $$\rho$$ has a Schottky decomposition; see [15, Part B, Theorem 11.2.4]. In genus $$g=2$$, there exist nonelementary representations $$\rho \in \mathrm{Hom}(\Gamma , \mathrm{PSL}_2 (\mathbb {C}))$$ that do not admit a Schottky decomposition: the pentagon representations , see [26]. In the case where $$\rho$$ is a pentagon representation, there exists a branched projective structure on $$\Sigma _2$$ with a single conical point of total angle $$2\pi (1 + 1)$$ and holonomy $$\rho$$, see [26, Theorem 1.6].…”

“…In genus $$g=2$$, there exist nonelementary representations $$\rho \in \mathrm{Hom}(\Gamma , \mathrm{PSL}_2 (\mathbb {C}))$$ that do not admit a Schottky decomposition: the pentagon representations , see [26]. In the case where $$\rho$$ is a pentagon representation, there exists a branched projective structure on $$\Sigma _2$$ with a single conical point of total angle $$2\pi (1 + 1)$$ and holonomy $$\rho$$, see [26, Theorem 1.6]. Moreover, we can assume that there exists a curve based at the conical point that develops injectively, see the red curve in [26, figure 8], whose exponential develops injectively.…”

“…It is also known since Poincaré [32], see also [28], that the restriction of $$\mathrm{Hol}$$ to the subset of $$\mathcal {P}$$ of projective structures with a fixed underlying complex structure is injective. A celebrated theorem of Gallo, Kapovich, and Marden [15] characterizes the image of $$\mathrm{Hol}_{|\mathcal {P}}$$, see also [26]. Namely, they show that if $$g \geqslant 2$$, the image of $$\mathcal {P}$$ by $$\mathrm{Hol}$$ is the set of nonelementary representations that lift to $$\mathrm{SL}_2 (\mathbb {C})$$.…”