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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 \(\mathbb{R}^n\) 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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