  We construct infinitely many linearly independent quasi-homomorphisms on the
mapping class group of a Riemann surface with genus at least two which vanish
on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov
on bounded width in Heegaard splittings. Another corollary is that there are
infinitely many linearly independent quasi-invariants on the Heegaard
splittings of three-manifolds.
