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Fundamental groups of blow-ups. (English) Zbl 1080.52512
Summary: Many examples of nonpositively curved closed manifolds arise as real blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group $W$ and if the blow-up locus is $W$-invariant, then the resulting manifold will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope $P$. In other words, $M$ admits a tiling with tile $P$. The universal covers of such examples yield tilings of ${ℝ}^{n}$ whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these “mock reflection groups", and develop a theory of tilings that includes the examples coming from blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile $P$ is either the permutohedron or the associahedron.

##### MSC:
 52C35 Arrangements of points, flats, hyperplanes 52C22 Tilings in $n$ dimensions (discrete geometry)