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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.

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    $\begingroup$ I'm pretty sure the answer is yes: you just have to make sure you model $M$ on a manifold with corners. Then the reflections should fit together to get a smooth manifold structure on M'. $\endgroup$
    – Ian Agol
    Mar 14, 2011 at 19:31
  • $\begingroup$ @Ian: Yes, that is what I thought too. But I need something to refer to. It is needed for my paper. $\endgroup$
    – user6976
    Mar 14, 2011 at 19:50
  • $\begingroup$ @Ian: Davis also writes that in dim 4 the situation is different. $\endgroup$
    – user6976
    Mar 14, 2011 at 19:58
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    $\begingroup$ Actually, the book is on his homepage: math.osu.edu/~mdavis/my%20book/My%20book.html $\endgroup$ Mar 14, 2011 at 21:55
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    $\begingroup$ A slightly pedantic comment: the group $G$ in Mark's posting does not come equipped with an action on $\mathbb{R}^5$. The manifold $M'$ with the proper $G$-action is constructed from $G\times M$, i.e., the direct product of $G$ and $M$, by identifying pieces of the various copies of $\partial M$. Provided that the 2-complex $C$ that you started with was aspherical, the universal cover of $M'$ will be contractible. But even in this case, it need not be homeomorphic to $\mathbb{R}^5$. $\endgroup$
    – IJL
    Jun 7, 2018 at 13:38

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