Hitchin components for orbifolds

• Applicants: Daniele Alessandrini, Ph.D.; Dr. Gye-Seon Lee
• Subject Area: Mathematics
• Term: from 2017 to 2020
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 340014145

Teichmüller spaces parametrise the set of hyperbolic structures on a surface. They can also be seen as spaces of discrete and faithful representations of the fundamental group of the surface in PSL(2,R). These representations are called Fuchsian representations.Higher Teichmüller Theory tries to generalise the theory of Teichmüller spaces replacing the group PSL(2,R) with another Lie group G. In this project G will be usually PSL(n,R). Hitchin representations are a special kind of representations of the surface group in PSL(n,R), having similar properties to the Fuchsian representations. The sets of such representations are called Hitchin components, and they generalise Teichmüller spaces.In this project the study of Hitchin components will be generalised to a more general family of groups, namely the fundamental groups of 2-dimensional orbifolds. This is a large family of groups including for example 2-dimensional hyperbolic Coxeter groups. The case of PSL(2,R) was already studied by Thurston, who introduced hyperbolic structures on orbifolds and studied their Teichmüller spaces. The case of PSL(3,R) was studied by Choi and Goldman. This project is about the general case of Hitchin components of orbifold groups for PSL(n,R). We already proved some preliminary results, the Hitchin components are homeomorphic to a vector space, and we have a formula for their dimensions.This study will be further developed to get a better understanding of these Hitchin components. The most interesting cases are the cases of "small" orbifolds, some orbifolds whose Hitchin components have very small dimensions, say 0, 1 or 2. We can classify all the Hitchin components of dimension 0, this is very interesting because it gives surprising examples of rigidity. Some applications to the deformation spaces of real projective structures on 3-manifolds are given, we can find examples of Seifert fibered 3-manifolds admitting a rigid real projective structure.The case of Hitchin components of dimension 1 is also very interesting, because the geometry of these components can be easy to understand. Moreover, they are a very good candidate to study phenomena of degeneration: what happens when a sequence of representations diverges?Moreover, geometric coordinates on these Hitchin components for orbifolds will be developed, a suitable generalization of the Fock-Goncharov coordinates. Having explicit coordinates can be very useful to understand these deformation spaces.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity