  We prove that for any V>0, there exist a hyperbolic manifold M_V, so that
Vol(M_V) < 2.03 and LinVol(M_V) > V.
  The proof requires study of cosmetic surgery on links (equivalently, fillings
of manifolds with boundary tori). There is no bound on the number of components
of the link (or boundary components). For statements, see the second part of
the introduction. Here are two examples of the results we obtain:
  1) Let K be a component of a link L in S^3. Then "most" slopes on K cannot be
completed to a cosmetic surgery on L, unless K becomes a component of a Hopf
link.
  2) Let X be a manifold and \epsilon>0. Then all but finitely many hyperbolic
manifolds obtained by filling X admit a geodesic shorter than \epsilon\ (note
that this finite set may correspond to an infinitely many fillings).
