  Let F be R or C, d the dimension of F over R. 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 the (2d-1)-sphere, 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; the proof is a direct construction, using ``Legendrian
inscription''. In contrast, if the (affine or hyperbolic) R-line arrangement L
is connected, then its complexification (an affine or hyperbolic C-line
arrangement) has a fibered link at infinity; the proof uses A'Campo's divides.
