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 $S_g$ 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 $k^{\rm th}$ term of its lower central series, $k \geq 1$. 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 ${\cal I}(S_g)$, we prove that $L({\cal I}(S_g))$, the logarithm of the minimal dilatation in ${\cal I}(S_g)$, satisfies $.197 < L({\cal I}(S_g))< 4.127$. 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 $g\to\infty$.