Hyperbolic 3-manifolds fibering over the circle

Question

Suppose you have the mapping torus $M_\phi$ of some pseudo-Anosov map $\phi.$ Is there some sufficient or necessary condition on $\phi$ to assure that $M_\phi$ has large injectivity radius? I am aware of Jeff Brock et al's results on volume bounds, but this is not quite the same...

Answer 1

A related question is studied by Minsky in the paper "Bounded geometry for Kleinian groups": http://arxiv.org/abs/arXiv:math/0105078

The main theorem gives a necessary and sufficient condition for the infinite cyclic cover $\tilde{M}_{\phi}$ of a pseudo-Anosov mapping torus to have large injectivity radius, in terms of the geometry of the action of $\phi$ on the curve complex $\mathcal{C}(S)$, for $S$ the fiber of your mapping torus.

It states that given $\epsilon>0$, there is some $K>0$ depending on $\epsilon$ and $S$, such that if $\tilde{M}_{\phi}$ has injectivity radius at least $\epsilon$, then all proper subsurface projections satisfy

$$ d_{Y}(\nu^{-}, \nu^{+}) <K. $$

Here, $Y \subset S$ is an essential, properly embedded subsurface, and $\nu^{-}, \nu^{+}$ are the stable and unstable laminations for $\phi$.

This condition is also sufficient (in the sense that given any $K'>0$, there exists some $\epsilon'$ such that if all subsurface projections are uniformly bounded above by $K'$, then your degenerate surface group $\tilde{M}_{\phi}$ has injectivity radius at least $\epsilon'$). In the literature, any pseudo-Anosov $\phi$ satisfying the subsurface projection criterion above for $K$ is said to have "$K$-bounded combinatorics".

This result says something about the injectivity radius of the mapping torus $M_{\phi}$, so long as the loop corresponding to the circle direction (i.e., the generator of the infinite cyclic group of deck transformations) isn't too short. But using work of Brock-Bromberg (see Section 7 of the paper "Geometric inflexibility and 3-manifolds that fiber over the circle: http://www.math.brown.edu/~brock/home/text/papers/inflex/www/inflex.pdf), and also Minsky, we can ensure that this doesn't happen by taking a large power of $\phi$.

The relevant result is that given a pseudo-Anosov $\phi$, its stable translation length in $\mathcal{C}(S)$ is coarsely equal to "thick distance" in $M_{\phi}$, which is the length of the shortest arc traversing the circle direction of $M_{\phi}$, after electrifying all thin regions (adding in an extra point $x$ and declaring all Margulis tubes of $M_{\phi}$ to be within $1/2$ of $x$).

So to summarize, here is a necessary condition on $\phi$ for $M_{\phi}$ to be $\epsilon$-thick:

$\phi$ must have $K(\epsilon,S)$-bounded combinatorics.

The problem is that, at least to my knowledge, none of this is effective. Given $S$ and $\epsilon$, it's completely unknown how small one needs to take $K$ (besides extremely soft statements, such that it's impossible for $K$ to be uniform in either $\epsilon$ or the topological complexity of $S$).

On the other hand, If you choose $\phi$ to have large stable translation length, and $K'$-bounded combinatorics, these results tell you that there is some $\epsilon'$ such that $M_{\phi}$ is $\epsilon'$-thick, but again it's difficult to estimate $\epsilon'$ in terms of $K'$ and $S$.

By the way, another way of thinking about all of this is that large injectivity radius of $\tilde{M}_{\phi}$ is implied by the Teichmuller geodesic connecting the end invariants $\nu^{+}, \nu^{-}$ residing in some sufficiently thick part of Teichmuller space (this interpretation is implied by the work referenced above, and also Rafi's work- see "A characterization of short curves along a Teichmuller geodesic": http://www.math.toronto.edu/~rafi/Papers/Short-Curves.pdf).