Low dilatation pseudo-Anosovs on punctured surfaces and volume.

Abstract

For a pseudo-Anosov homeomorphism f on a closed surface of genus \(g\ge 2\), for which the entropy is on the order \(\frac{1}{g}\) (the lowest possible order), Farb-Leininger-Margalit showed that the volume of the mapping torus is bounded, independent of g. We show that the analogous result fails for a surface of fixed genus g with n punctures, by constructing pseudo-Anosov homeomorphism with entropy of the minimal order \(\frac{\log n}{n}\), and volume tending to infinity.