Transitivity of Surface Dynamics Lifted to Abelian Covers

A homeomorphism f of a manifold M is called H1-transitive if there is a transitive lift of an iterate of f to the universal Abelian cover M̃ . Roughly speaking, this means that f has orbits which repeatedly and densely explore all elements of H1(M). For a rel pseudo-Anosov map φ of a compact surface M we show that the following are equivalent: (a) φ is H1-transitive, (b) the action of φ on H1(M) has spectral radius one, and (c) the lifts of the invariant foliations of φ to M̃ have dense leaves. The proof relies on a characterization of transitivity for twisted Z-extensions of a transitive subshift of finite type.