Suppose you have the mapping torus $M_\phi$ of some pseudoAnosov 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...
1 Answer
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 pseudoAnosov 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 pseudoAnosov $\phi$ 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 BrockBromberg (see Section 7 of the paper "Geometric inflexibility and 3manifolds 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 pseudoAnosov $\phi$, 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/ShortCurves.pdf).

$\begingroup$ By the way, the homotopically nontrivial loops in a fiber bundle have length bounded below uniformly in terms of the genus. However, if you want to get injectivity radius above a certain threshold, then I agree one should take a power or choose large enough translation length (although I don't think Minsky's theorem would be able to guarantee large injectivity radius for large genus, i.e. the $\epsilon'$ might be small). $\endgroup$– Ian AgolDec 15, 2013 at 19:21

$\begingroup$ @IanAgol I am actually happy with bounds depending on the genus, as long as I can meet the injectivity radius constraint by putting some (reasonable) conditions on the map. $\endgroup$ Dec 15, 2013 at 19:30