An upper bound on the asymptotic translation lengths on the curve graph and fibered faces

Abstract

We study the asymptotic behavior of the asymptotic translation lengths on the curve complexes of pseudo-Anosov monodromies in a fibered cone of a fibered hyperbolic 3-manifold $M$ with $b_1(M)\ge 2$. For a sequence $({\Sigma }_n,{\psi }_n)$ of fibers and monodromies in the fibered cone, we show that the asymptotic translation length on the curve complex is bounded above by $1/\chi ({\Sigma }_n{)}^{1+1/r}$ as long as their projections to the fibered face converge to a point in the interior, where $r$ is the dimension of the ${\psi }_n$-invariant homology of ${\Sigma }_n$ (which is independent of $n$). As a corollary, if $b_1(M)=2$, the asymptotic translation length on the curve complex of such a sequence of primitive elements behaves like $1/\chi ({\Sigma }_n{)}^2$. Furthermore, together with a work of E. Hironaka, our theorem can be used to determine the asymptotic behavior of the minimal translation lengths of handlebody mapping class groups and the set of mapping classes with homological dilatation one.

13 pages, 1 figure. The proofs for Prop. 4, Prop. 6, and Lemma 8 have been expanded. Other minor changes were made to incorporate referee's comments