## The lower central series and pseudo-Anosov dilatations

Authors:

Benson Farb,

Christopher J. Leininger,

Dan Margalit

Benson Farb,

Christopher J. Leininger,

Dan Margalit

Comments: 26 pages, 6 figures

Subj-class: Geometric Topology; Dynamical Systems

MSC-class: 37E30 (Primary) 57M60, 37B40 (Secondary)

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.

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