Pseudo-anosov Homeomorphisms and the Lower Central Series of a Surface Group

Denote by Mod(S) the mapping class group of a compact, oriented surface S = Sg,1 of genus g ≥ 2 with one boundary component; i.e. Mod(S) is the group of homeomorphisms of S fixing ∂S pointwise up to isotopies fixing ∂S pointwise. A basic question to contemplate is: what topological or dynamical data of a mapping class can be extracted from various kinds of algebraic data? Since pseudo-Anosovs are the more complex mapping classes topologically and dynamically, we would like to know if a given mapping class is pseudo-Anosov; i.e. it has a representative homeomorphism which leaves invariant a pair of transverse measured foliations. One kind of algebraic data is the action of a mapping class on Γ := π1(S, ∗) and its various quotients. Specifically, consider the sequence of k-step nilpotent quotients Nk := Γ/Γk+1 where {Γk} is the lower central series of Γ defined inductively by Γ1 = Γ Γk = [Γ,Γk−1] for k > 1