Pentagon representations and complex projective structures on closed surfaces

TL;DR

This paper introduces pentagon representations of surface groups into PSL(2,C), characterizes those without Schottky decompositions, and corrects a gap in the proof that all non-elementary representations are holonomies of complex projective structures.

Contribution

It defines pentagon representations, characterizes non-Schottky representations, and repairs a gap in the proof relating surface group representations to complex projective structures.

Findings

Pentagon representations are exactly the non-elementary, non-Schottky surface group representations.

Pentagon representations can be realized as holonomies of complex projective structures.

The proof that all non-elementary representations are holonomies is corrected and completed.

Abstract

We define a class of representations of the fundamental group of a closed surface of genus $2$ to $\mathrm{PSL}_2 (\mathbb C)$: the pentagon representations. We show that they are exactly the non-elementary $\mathrm{PSL}_2 (\mathbb C)$-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 $\mathrm{PSL}_2 (\mathbb C)$ 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.