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

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 \mathbbH^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(\mathbbR) up to conjugation is disconnected.