Some fibered and non-fibered links at infinity of hyperbolic complex line arrangements

Let F be R or C, d:=dimR(F). Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K∞(L) comprising k unknotted (d−1)-spheres in S2d−1, whose topology reflects the combinatorics of L “at infinity”. The class of links at infinity of affine F-line arrangements is properly included in the class of links at infinity of hyperbolic F-line arrangements. Many links at infinity of (essentially non-affine) connected hyperbolic C-line arrangements are far from being fibered. In contrast, if the (affine or hyperbolic) R-line arrangement LR⊂P(R) is connected, and L=C LR⊂P(C) is its complexification, then K∞(L) is fibered.