Transitivity of Surface Dynamics Lifted to Abelian Covers

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.