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

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 [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.

arXiv comment32 pages, to appear in Transactions of the American Mathematical Society