On the symplectic geometry of branched hyperbolic surfaces in genus two

TL;DR

This paper develops Fenchel-Nielsen-like coordinates for branched hyperbolic structures on genus-2 surfaces, demonstrating their symplectic properties and introducing bow-tie representations to parametrize most holonomies.

Contribution

It constructs new coordinate systems for branched hyperbolic structures on genus-2 surfaces and establishes their symplectic properties, extending Fenchel-Nielsen coordinates to this setting.

Findings

Coordinates satisfy Wolpert's formula

Bow-tie representations describe most holonomies

Le Fils' pentagon representations lie outside the bow-tie locus

Abstract

We construct analogues of Fenchel-Nielsen coordinates on an open and dense subset of the space of holonomies of branched hyperbolic structures on a closed genus-2 surface. We show that these coordinates satisfy an analogue of Wolpert's magic formula, and thus provide Darboux charts for the Goldman symplectic form. To this end, we revisit the parametrization of hyperbolic structures on a one-holed torus and describe a simple polygonal model that makes both length and twist parameters transparent. Gluing two such polygons leads to the notion of bow-tie representations of a genus-2 surface group. We prove that bow-tie representations account for most holonomies of branched hyperbolic structures, though not all: for example, Le Fils' pentagon representations form a real codimension-2 family of holonomies lying outside the bow-tie locus.