Given a pseudo Anosov mapping class $f:S_{g,n}\rightarrow S_{g,n}$ is the Lefschetz number for $f^m$ negative for some $m$ depending only on $(g,n)$?

If the leading eigenvalue for the action on homology is 1 then we are done. If the leading eigenvalue $\lambda\neq1$ then the sequence of Lefschetz numbers is dominated $\lambda^m$ and so the Lefschetz number will eventually be negative.

Thanks!