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

TL;DR

This paper constructs examples of anti-de Sitter groups that are GHC-regular but not quasi-isometric to hyperbolic or symmetric spaces, challenging existing conjectures and providing new insights into Coxeter groups.

Contribution

It provides explicit examples of GHC-regular groups in AdS spaces that are not lattices and are not quasi-isometric to hyperbolic or symmetric spaces, disproving prior conjectures.

Findings

Existence of GHC-regular groups not quasi-isometric to hyperbolic spaces

Disproof of a specific conjecture in the literature

Disconnected spaces of GHC-regular representations for certain Coxeter groups

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-M\'erigot [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-Gu\'eritaud-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.