The symplectic geometry of branched hyperbolic structures

Arnaud Maret
Strasbourg

Abstract

Hyperbolic geometry is one of the most iconic geometries studied on surfaces. When conical singularities at certain points are allowed, whose angles are integer multiples of 2π, one obtains branched hyperbolic structures. An important problem, proposed by Goldman, is to determine which surface group representations arise as holonomies of branched hyperbolic structures. In this talk, I will focus on the case of genus-2 surfaces and explain how to construct Fenchel--Nielsen (i.e. Darboux) coordinates on the space of holonomies of these branched hyperbolic structures. This is joint work with Gianluca Faraco.