Special cubulation of strict hyperbolization (Q6536649)

The hyperbolization procedure of \textit{M. W. Davis} and \textit{T. Januszkiewicz} [J. Differ. Geom. 34, No. 2, 347--386 (1991; Zbl 0723.57017)] and strict hyperbolization procedure of \textit{R. M. Charney} and \textit{M. W. Davis} [Topology 34, No. 2, 329--350 (1995; Zbl 0826.53040)] are methods for producing locally \(\mathrmCAT(0)\) or \(\mathrmCAT(-1)\) spaces out of simplicial complexes. If the input simplicial complex is a manifold, then the output in both constructions will also be a manifold. Thus, hyperbolization procedures can be applied in great generality to produce examples of aspherical n-manifolds with interesting properties. Recently, \textit{P. Ontaneda} [Publ. Math., Inst. Hautes Étud. Sci. 131, 1--72 (2020; Zbl 1442.53026)] showed that, provided the input manifold is smooth, the output manifold can be chosen so that it admits a Riemannian metric of negative sectional curvature.\N\NIn this paper, the authors show that many of the groups that one can obtain by strict hyperbolization are co-compactly cubulated (Theorem 1.1) and moreover virtually compact special (Theorem 1.2), and hence, amongst other things, are residually finite, virtually RFRS, virtually bi-orderable and admit Anosov representations.\N\NThe Charney-Davis strict hyperbolization procedure involves as an input an arithmetic real hyperbolic lattice \(\Gamma\). The authors produce an action of the output group \(G\) on a \(\mathrmCAT(0)\) cube complex whose cell-stabilisers are quasiconvex in both \(G\) and \(\Gamma\). The group \(\Gamma\) is known to be virtually compact special by work of \textit{N. Bergeron} et al. [J. Lond. Math. Soc., II. Ser. 83, No. 2, 431--448 (2011; Zbl 1236.57021)]. The authors are then able to deduce that \(G\) is virtually compact special by applying a result of \textit{D. Groves} and \textit{J. F. Manning} [Geom. Topol. 27, No. 9, 3387--3460 (2023; Zbl 1528.20067), Theorem D].