Anti-de Sitter strictly GHC-regular groups which are not lattices

Lee, Gye-Seon; Marquis, Ludovic

doi:10.1090/tran/7530

Anti-de Sitter strictly GHC-regular groups which are not lattices

Authors: Gye-Seon Lee and Ludovic Marquis
Journal: Trans. Amer. Math. Soc. 372 (2019), 153-186
MSC (2010): Primary 20F55, 20F65, 20H10, 22E40, 51F15, 53C50, 57M50, 57S30
DOI: https://doi.org/10.1090/tran/7530
Published electronically: April 4, 2019
MathSciNet review: 3968766
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Abstract: For , 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 a work of Barbot et al. and disproves Conjecture 8.11 of Barbot-Mérigot [Groups Geom. Dyn. 6 (2012), pp. 441-483].

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 (Ph.D. Thesis) for Coxeter groups built on Danciger-Guéritaud-Kassel's 2017 work and find examples of Coxeter groups such that the space of strictly GHC-regular representations of into $\mathrm {PO}_{d,2}(\mathbb{R})$ up to conjugation is disconnected.

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