3 Made notation in last paragraph match notation in previous.

Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudoanosov pseudo-Anosov map $\phi$ in a surface $S$. Is there anyway of estimating any way to estimate the length of the geodesic representing a given curve in the surface in terms of the map $\phi$, that \phi$? That is, knowing something like the stable and unstable foliations for the map or something equivalent, can you estimate the length of a given curve? Any references for something like this are really appreciated. For example, if you take a mapping torus$M_{\phi}$, drill one simple nontrivial curve$\alpha$in the surface and make re-glue by$\sigma^n$, a large dehn Dehn twist , about$\alpha$, you are going to get a hyperbolic mapping torus$M_{\phi\sigma^{n}}$, where$\sigma$is a dehn twist in the curve, in M_{\phi\sigma^{n}}$. In this manifold $\alpha$ is going to be very short.

Another example, if you take a map $\psi = \phi^n\sigma$ phi\sigma^n$where$\sigma$\phi$ is pseudo-anosov in all of $S$ and $\phi$ \sigma$is a pseudoanosov pseudo-Anosov just in a subsurface of$S$, X \subset S$, I think the curves in the complement of the subsurface $X$ have to be very small for $n$ large, right?

2 edited title

1

# Pseudoanosov mapping torus and injectivity radius.

Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudoanosov map $\phi$ in a surface $S$. Is there anyway of estimating the length of the geodesic representing a given curve in the surface in terms of the map $\phi$, that is, knowing something like the stable and unstable foliations for the map or something equivalent, can you estimate the length of a given curve? Any references for something like this are really appreciated.

For example, if you take a mapping torus $M_{\phi}$, drill one simple nontrivial curve $\alpha$ in the surface and make a large dehn twist, you are going to get a hyperbolic mapping torus $M_{\phi\sigma^{n}}$, where $\sigma$ is a dehn twist in the curve, in this manifold $\alpha$ is going to be very short.

Another example, if you take a map $\psi = \phi^n\sigma$ where $\sigma$ is pseudo-anosov in all of $S$ and $\phi$ is a pseudoanosov just in a subsurface of $S$, I think the curves in the complement of the subsurface have to be very small for $n$ large, right?