Degrees of Stretch Factors and Counterexamples to Farb's Conjecture

Abstract

Franks and Rykken proved that any pseudo-Anosov map with quadratic stretch factor and orientable invariant foliations, can be obtained by lifting a torus map by a branched or unbranched covering map. Farb conjectured that this generalizes to higher degree stretch factors. We construct counterxamples to this conjecture. The proof relies on studying the Perron-Frobenius degrees of our constructed stretch factors and a careful analysis of their Galois conjugates.