We consider pseudo-Anosov mapping classes on a closed orientable surface of genus $g$ that fix a rank $k$ subgroup of the first homology of the surface. We first show that there exists a uniform constant $C>0$ so that the minimal asymptotic translation length on the curve complex among such pseudo-Anosovs is bounded below by $C over g(2g-k+1)$. This interpolates between results of Gadre-Tsai and of the first author and Shin, who treated the cases of the entire mapping class group ($k = 0$) and the Torelli subgroup ($k = 2g$), respectively. We also discuss possible strategy to obtain an upper bound. Finally, we construct a pseudo-Anosov on a genus $g$ surface whose maximal invariant subspace is of rank $2g-1$ and the asymptotic translation length is of $asymp 1/g$ for all $g$. Such pseudo-Anosovs are further shown to be unable to normally generate the whole mapping class groups. As Lanier-Margalit proved that pseudo-Anosovs with small translation lengths on the Teichmuller spaces normally generate mapping class groups, our observation provides a restriction on how small the asymptotic translation lengths on curve complexes should be if the similar phenomenon holds for curve complexes.
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