Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length of homotopically non-trivial loops in $M(f)$ bounded below in terms of the genus $g$ of the fiber?
A comment to this question Hyperbolic 3-manifolds fibering over the circle suggests that this is the case. Any details and references are highly welcome.
