Recall that a complex hyperplane complement is a topological space obtained by removing a collection of complex codimensional one affine hyperplanes from C^n (or a convex cone of C^n). The fundamental groups of these spaces contain many interesting examples, including pure braid groups and (pure) right-angled Artin groups, though in general the fundamental groups of complex hyperplane complements are far from being understood. We showed that some classes of these fundamental groups can be equivariantly ``thickened'' to a metric space which are non-positively curved in an appropriate sense, and discuss several algorithmic and , geometric and topological consequences of such non-positive curvature condition. This is joint work with D. Osajda.