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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Anti-de Sitter strictly GHC-regular groups which are not lattices
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by Gye-Seon Lee and Ludovic Marquis PDF
Trans. Amer. Math. Soc. 372 (2019), 153-186 Request permission

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
  • MR Author ID: 832440
  • Email: lee@mathi.uni-heidelberg.de
  • Ludovic Marquis
  • Affiliation: Université de Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France
  • MR Author ID: 906548
  • Email: ludovic.marquis@univ-rennes1.fr
  • 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).
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 3968766