The lower central series and pseudo-Anosov dilatations

TL;DR

This paper investigates the relationship between algebraic complexity and dynamical complexity in pseudo-Anosov homeomorphisms, showing that certain algebraic constraints lead to bounded dilatations independent of genus, contrasting with Penner's asymptotics.

Contribution

It establishes bounds on minimal dilatations for pseudo-Anosov homeomorphisms acting trivially on quotients of the lower central series, revealing new asymptotic behaviors.

Findings

Minimal dilatation bounds are independent of genus g.

Dilatations tend to infinity with the lower central series index k.

Some pseudo-Anosov classes have translation lengths tending to zero as g increases.

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'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.