The geometry of Deroin-Tholozan representations
Aaron Fenyes, Arnaud Maret
TL;DR
This work provides a concrete geometric realization of Deroin--Tholozan representations as holonomies of hyperbolic cone spheres by assembling simple building blocks called samosas (and hamantashe) into chained pants assemblies. The authors develop a parameterization of hyperbolic cone structures via triangulations, establish injection and (under a smooth-atlas conjecture) diffeomorphism properties for the realization map, and show that any DT representation arises from a hamantash assembly, yielding a geometric interpretation of action-angle coordinates. By unfolding samosas into hyperbolic polygons, they connect the assembly data to triangle-chain data and prove that the holonomies match the DT data, providing a surjective bridge from cone-sphere constructions to DT representations. The approach gives a versatile framework to explore deformations of cone structures while controlling holonomy, and it opens avenues to relate DT coordinates to Teichmüller-type parameters through samosa-based realizations.
Abstract
We present a way to build hyperbolic spheres with conical singularities by gluing together simple building blocks. Our construction provides good control over the holonomy of the resulting hyperbolic cone sphere. In particular, it can be used to realize any Deroin-Tholozan (DT) representation as the holonomy of a hyperbolic cone sphere. Our construction is inspired by the correspondence between DT representations and chains of triangles in the hyperbolic plane. It gives a geometric interpretation of certain action-angle coordinates on the space of DT representations, which come from this correspondence.