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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
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

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