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Hitchin components for orbifolds

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):
1Hitchin components for orbifolds

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## Team Members

Dr. Daniele Alessandrini
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