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 CP2 with singularities at a collection of complex lines. In the former case we conjecture that quotient spaces always have a CAT ramification and prove this in several cases. In the latter case we prove that the ramification is CAT if the metric on CP2 is non-negatively curved. We deduce that complex line arrangements in CP2 studied by Hirzebruch have aspherical complement.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory