Pseudo-Anosov map, Heegaard splitting, hyperbolic 3-manfold

Question

Hi, I am interested in the relationship between the pseudo-anosov map and volume of the hyperbolic 3-manifold.

Assume $H_{1}$ and $H_{2}$ are two handlebodies with $\partial H_{1}=\partial H_{2}=S$.

Question 1:For any pseudo-anosov homeomorphism $\psi: S\rightarrow S$, if the $n\in N$ is large enough, is $M_{\psi^{n}}=H_{1}\cup_{\psi^{n}} H_{2}$ hyperbolic?

Question 2:Given a pseudo-anosov map $\psi$, suppose $M_{\psi^{n}}=H_{1}\cup_{\psi^{n}} H_{2}$ is hyperbolic, for any $n\geq k$, where $k\in N$. How does the Vol(M) change when $n$ goes to infinity?

Answer 1

For Question 1:

Souto and Namazi (pdf link) showed that for a generic pseudo-anosov homeomorphism $\psi$ and $\epsilon >0$, there is $n_\epsilon$ such that $M_{\psi^n}$ admits a Riemannian metric with all sectional curvatures between $-1-\epsilon$ and $-1+\epsilon$ for all $n\ge n_\epsilon$.
Namazi (pdf link) used a theorem of Tian to show that for $\epsilon$ small enough these manifolds are actually hyperbolic.

For Question 2:

The manifolds Souto and Namazi construct have injectivity radius bounded below independently of $n$ and $\epsilon$, so the volumes must grow without bound as $n$ goes to infinity.

Answer 2

For Question 1, the answer is no. In fact, there are pseudo-Anosov maps $\psi$ that preserve a handlebody $H_1$ (the existence of such a map follows from work of Masur-Minsky, although probably appears earlier). $H_1\cup_{\psi^n} H_2 = H_1\cup_S H_2$, so if $H_1\cup_S H_2$ is not hyperbolic, then neither will $M_{\psi^n}$.

For Question 2, the answer is a bit more complicated. Namazi-Souto prove if $\psi$ is "generic", which means that the stable lamination $\lambda_+$ of $\psi$ is not a limit of meridians of $H_2$, and the unstable lamination $\lambda_-$ of $\psi$ is not a limit of meridians of $H_1$, then $M_{\psi^n}=H_1\cup_{\psi^n} H_2$ is hyperbolic for large $n$. Moreover, for any $\epsilon >0$ and large enough $n$, there are metrics on $M_{\psi^n}$ with curvatures pinched between $-1-\epsilon$ and $-1+\epsilon$, with volume growing linear with $n$. This implies that the hyperbolic volume of $M_{\psi^n}$ grows linearly as well, either by applying volume comparison theorems of Besson-Courtois-Gallot, or by an unpublished preprint of Tian.

If the map $\psi$ is not generic, (say $\lambda_+$ is a limit of meridians of $H_2$), then Biringer-Johnson-Minsky prove that a power of $\psi$ extends over a compression body inside of $H_2$. This does not necessarily imply that the manifolds $M_{\psi^n}$ is not hyperbolic, but if they are I'm not sure how fast the volume grows; I suspect it would still grow linearly if some power does not extend entirely over $H_1$ or $H_2$ (like in the answer to Question 1).