The Lower Central Series and Pseudo-Anosov Dilatations

The theme of this paper is that algebraic complexity implies dynamical complexity for pseudo-Anosov homeomorphisms of a closed surface${\rm{S}}_{\rm{g}}$of genus g. Penner proved that the logarithm of the minimal dilatation for a pseudo-Anosov homeomorphism of${\rm{S}}_{\rm{g}}$tends to zero at the rate 1/g. We consider here the smallest dilatation of any pseudo-Anosov homeomorphism of${\rm{S}}_{\rm{g}}$acting trivially on$\Gamma /\Gamma _k $, the quotient of$\Gamma \, = \,\pi _1 (S_g )$by the${\rm{K}}^{{\rm{th}}}$term of its lower central series, k > 1. In contrast to Penner's asymptotics, we prove that this minimal dilatation is bounded above and below, independently of g, with bounds tending to infinity with k. For example, in the case of the Torelli group${\rm{I(S}}_g )$, we prove that${\rm{L(I(S}}_g ))$, the logarithm of the minimal dilatation in${\rm{I(S}}_g )$, satisfies .197