The project belongs to the research area called Higher Teichmüller Theory. Given a closed surface \(S\) of genus \(g\ge 2\), and a reductive Lie group , we consider the *character variety* of \(\pi_1(S)\) in \(G\), the space of representations of the fundamental group of \(S\) into \(G\), up to the action of \(G\) by conjugation. The more precise definition:

\(X(\pi_1(S),G)=Hom^+( \pi_1(S),G)/G\)

where \(Hom^+\)denotes the space of reductive representations of \(\pi_1(S)\) in \(G\), a subset where the action of \(G\) by conjugation has a good quotient. The same space can also be seen as a parameter space of equivalence classes of vector bundles with flat connections, or of local systems.

Our preliminary work concerns:

- Compactification of Hitchin component and character varieties
- Geometric structures and Higgs bundles
- Teichmüller space for surfaces of infinite types
- Collar lemma for Hitchin representations
- Convex projective structures on non-hyperbolic three-manifolds

The work programme for this project includes:

- Define Hitchin components for orbifolds and analyze its topological and geometric properties
- Study the degeneration of Hitchin representations for small orbifolds, to understand general surface groups
- Parametrize Hitchin components for orbifolds

## Publications

For d = 4, 5, 6, 7, 8, we exhibit examples of \(\mathrm{AdS}^{d,1}\) strictly GHC-regular groups which are not quasi-isometric to the hyperbolic space \(\mathbb{H}^d\), nor to any symmetric space. This provides a negative answer to Question 5.2 in [9A12] and disproves Conjecture 8.11 of Barbot-Mérigot [BM12]. We construct those examples using the Tits representation of well-chosen Coxeter groups. On the way, we give an alternative proof of Moussong's hyperbolicity criterion [Mou88] for Coxeter groups built on Danciger-Guéritaud-Kassel [DGK17] and find examples of Coxeter groups W such that the space of strictly GHC-regular representations of W into \(\mathrm{PO}_{d,2}(\mathbb{R})\) up to conjugation is disconnected.

**Related project(s):****1**Hitchin components for orbifolds

## Team Members

**Dr. Daniele Alessandrini**

Project leader

Ruprecht-Karls-Universität Heidelberg

daniele.alessandrini(at)gmail.com

**Dr. Shinpei Baba**

Researcher

Ruprecht-Karls-Universität Heidelberg

shinpei(at)mathi.uni-heidelberg.de

**Dr. Gye Seon Lee**

Project leader

Ruprecht-Karls-Universität Heidelberg

lee(at)mathi.uni-heidelberg.de

## Guests

**Prof. Dr. Florent Schaffhauser**

Universidad de los Andes

fm.schaffhauser416(at)uniandes.edu.co