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 ${\mathrm{PSL}}_2(\mathbb{C})$: 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.

19 pages, 8 figures