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Abstract

The complement of an arrangement of hyperplanes in $\mathbb C^n$ has a
natural bordification to a manifold with corners formed by removing (or "blowing up") tubular neighborhoods of the hyperplanes and certain of their intersections. When the arrangement is the complexification of a real simplicial arrangement, the bordification closely resembles Harvey's bordification of moduli space. We prove that the faces of the universal cover of the bordification are parameterized by the simplices of a simplicial complex
$\mathcal{C}$, the vertices of which are the irreducible "parabolic subgroups" of the fundamental group of the arrangement complement. So, the complex
$\mathcal{C}$ plays a similar role for an arrangement complement as the curve
complex does for moduli space. Also, in analogy with curve complexes and with
spherical buildings, we prove that $\mathcal{C}$ has the homotopy type of a
wedge of spheres.