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?



Pseudo-Anosov map, Heegaard splitting, hyperbolic 3-manfold
For Question 1:


Souto and Namazi ([pdf link](http://www.math.ubc.ca/~jsouto/papers/pseudo9-corrected.pdf)) 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](http://www.ma.utexas.edu/users/hossein/contents/main2.pdf)) 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.


