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.


52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
52C22 Tilings in \(n\) dimensions (aspects of discrete geometry)
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