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Transactions of the American Mathematical Society

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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
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Abstract: 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 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 $ 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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Additional Information

Gye-Seon Lee
Affiliation: Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Germany
Email: lee@mathi.uni-heidelberg.de

Ludovic Marquis
Affiliation: Université de Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
Email: ludovic.marquis@univ-rennes1.fr

DOI: https://doi.org/10.1090/tran/7530
Keywords: Anti-de Sitter spaces, Anosov representations, quasi-Fuchsian groups, Coxeter groups, discrete subgroups of Lie groups
Received by editor(s): August 30, 2017
Received by editor(s) in revised form: January 19, 2018, and February 9, 2018
Published electronically: April 4, 2019
Additional Notes: The first author was supported by the European Research Council under ERC-Consolidator Grant 614733 and by DFG grant LE 3901/1-1 within the Priority Programme SPP 2026 “Geometry at Infinity”, and he acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures and representation varieties” (the GEAR Network).
Article copyright: © Copyright 2019 American Mathematical Society