On the symplectic geometry of branched hyperbolic surfaces in genus two
Gianluca Faraco, Arnaud Maret
TL;DR
This work develops Fenchel–Nielsen–type coordinates for PSL(2,R) representations of genus-2 surface groups with Euler number ±1, focusing on two explicit families: pentagon representations and bow-tie representations. By introducing polygonal models for one-holed tori and assembling them into genus-2 structures, the authors produce Darboux charts for Goldman’s symplectic form on open dense subsets of the corresponding character components and establish Wolpert-type formulas. They show that pentagon and bow-tie representations are geometrizable as holonomies of branched hyperbolic structures, and describe their isomonodromic deformations via explicit geometric operations on the building blocks. The results provide a concrete, geometrically origin parametric framework for understanding the Goldman symplectic geometry on genus-2 moduli spaces and pave the way for extensions to broader classes of representations and higher genus. They also illuminate the relationship between hyperelliptic symmetry, polygonal decompositions, and the global structure of the moduli space, contributing to the broader program of geometrization of PSL(2,R) representations.
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 introduce 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.