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

Abstract

Let F be R or C , d:= dim R ( 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 S 2 d−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 L R ⊂ P( R) is connected, and L= C L R ⊂ P( C) is its complexification, then K ∞( L) is fibered.