## 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 $$\mathbb{R}^n$$ 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 (aspects of discrete geometry) 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry)
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### References:

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