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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

We extend the notion of Hitchin component from surface groups to orbifold groups and prove that this gives new examples of Higher Teichmüller spaces. We show that the Hitchin component of an orbifold group is homeomorphic to an open ball, and we compute its dimension explicitly. For example, the Hitchin component of the right-angled hyperbolic $$\ell$$-polygon reflection group into $$\mathrm{PGL}(2m,\mathbb{R})$$, resp. $$\mathrm{PGL}(2m+1,\mathbb{R})$$, is homeomorphic to an open ball of dimension $$(\ell-4)m^2+1$$, resp. $$(\ell-4)(m^2+m)$$. We also give applications to the study of the pressure metric and the deformation theory of real projective structures on 3-manifolds.

Related project(s):
1Hitchin components for orbifolds

Geometric structures on manifolds became popular when Thurston used them in his work on the Geometrization Conjecture. They were studied by many people and they play an important role in Higher Teichmüller Theory. Geometric structures on a manifold are closely related with representations of the fundamental group and with flat bundles. Higgs bundles can be very useful in describing flat bundles explicitly, via solutions of Hitchin's equations. Baraglia has shown in his Thesis that this technique can be used to construct geometric structures in interesting cases. Here we will survey some recent results in this direction, which are joint work with Qiongling Li.

Related project(s):
1Hitchin components for orbifolds

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