  A homeomorphism f of a manifold M is called H_1-transitive if there is a
transitive lift of an iterate of f to the universal Abelian cover \tM. Roughly
speaking, this means that f has orbits which repeatedly and densely explore all
elements of H_1(M). For a rel pseudo-Anosov map \phi of a compact surface M we
show that the following are equivalent: (a) \phi is H_1-transitive, (b) the
action of \phi on H_1(M) has spectral radius one, and (c) the lifts of the
invariant foliations of \phi to \tM have dense leaves. The proof relies on a
characterization of transitivity for twisted \Z^d-extensions of a transitive
subshift of finite type.
