King's College London

The ramification of a polyhedral space is defined as the metric completion of the universal cover of its regular locus. We consider mainly polyhedral spaces of two origins: quotients of Euclidean space by a discrete group of isometries and polyhedral metrics on (Formula presented.) with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a (Formula presented.) ramification and prove this in several cases. In the latter case we prove that the ramification is (Formula presented.) if the metric on (Formula presented.) is non-negatively curved. We deduce that complex line arrangements in (Formula presented.) studied by Hirzebruch have aspherical complement.