Anti-de Sitter strictly GHC-regular groups which are not lattices
Résumé
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.
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quasi_fuchsian_lorentz_Final.pdf (611.19 Ko)
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disconnected.pdf (148.94 Ko)
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disconnected.ps (1.51 Mo)
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