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Polyhedral complexes are obtained by gluing convex polytopes via isometric maps. In the talk I will explain how to use such complexes to construct complex-projective varieties with controlled singularities (normal crossings or ℤ/2-quotients of such) and with prescribed fundamental groups. The polyhedral complexes that appear in the construction come from geometric topology of 1960s (in the Euclidean case) and hyperbolic geometry (Dirichlet fundamental domains). This is joint work with Janos Kollar.