Davis' trick and Riemannian manifolds

2011-03-14T18:58:07Z

Let $C$ be a finite 2-dimensional aspherical complex. The Michael Davis' trick is basically the following: One embeds $C$ to, say, ${\mathbb R}^5$, then takes a regular neighborhood in ${\mathbb R}^5$, producing a manifold $M$ with boundary. Then we triangulate the boundary and using a right angled Coxeter reflection group $G$ acting on ${\mathbb R}^5$, by gluing together all the images $GM'$, produce a manifold $M'$ on which $G$ acts properly by isometries. Factoring by a torsion-free finite index subgroup $H < G$, produce a closed aspherical manifold $M'/H$ whose fundamental group contains $\pi_1(C)$.

Question. Can we get a closed smooth (Riemannian) aspherical manifold where $C$ $\pi_1$-embeds?

I think, by reading Davis' Annals paper published in 1983 that the answer is "yes", but I would like to make sure.

Davis' trick and Riemannian manifolds - MathOverflow

Davis' trick and Riemannian manifolds