The lower central series and pseudo-Anosov dilatations

Abstract

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface S_g of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of S_g tends to zero at the rate 1/g. We consider here the smallest dilatation of any pseudo-Anosov homeomorphism of S_g acting trivially on Gamma/Gamma_k, the quotient of Gamma = pi_1(S_g) by the k-th term of its lower central series, k > 0. In contrast to Penner&#39;s asymptotics, we prove that this minimal dilatation is bounded above and below, independently of g, with bounds tending to infinity with k. For example, in the case of the Torelli group I(S_g), we prove that L(I(S_g)), the logarithm of the minimal dilatation in I(S_g), satisfies .197 < L(I(S_g))< 4.127. In contrast, we find pseudo-Anosov mapping classes acting trivially on Gamma/Gamma_k whose asymptotic translation lengths on the complex of curves tend to 0 as g tends toward infinity.