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Let S be a closed orientable surface and let G be its fundamental group. We consider the smallest dilatation of any pseudo-Anosov homeomorphism of S acting trivially on G/G_k, the quotient of G by the k-th term of its lower central series, k > 0. We prove that this minimal dilatation is bounded above and below, independently of genus, with bounds tending to infinity with k. For example, in the case of the Torelli group I(S), we prove that L(I(S)), the logarithm of the minimal dilatation in I(S), satisfies .196 < L(I(S)) < 4.127. In contrast, we find pseudo-Anosov mapping classes acting trivially on G/G_k whose asymptotic translation lengths on the complex of curves tend to zero as genus tends to infinity.