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?


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.


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$M_{\psi^n}$ grows linearly as well, either by applying volume comparison theorems of Besson-Courtois-Gallot, or by an unpublished preprint of Tian.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).

  • 2
    $\begingroup$ One should also keep an eye out for the forthcoming paper of Brock and Souto, "Heegaard splittings, pants decompositions and volumes of hyperbolic 3-manifolds," where they (coarsely) obtain the volume for much more general Heegaard splittings. $\endgroup$ – Autumn Kent Sep 21 '12 at 12:55
  • $\begingroup$ @Richard: Do you have a copy of the paper? $\endgroup$ – yanqing Sep 24 '12 at 1:57
  • $\begingroup$ @yanqing: You might want to look at Section 8 of Souto's survey paper "Geometry, Heegaard splittings and rank of the fundamental group of hyperbolic 3-manifolds" at (arxiv.org/pdf/0904.0237.pdf) He sketches an argument obtaining bounds on volume in terms of the distance in the pants comlex of a strongly irreducible Heegaard surface. $\endgroup$ – b b Sep 26 '12 at 15:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.