Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations or conformal structures (the end invariants) on $S_1...S_n$ one is able to find a unique complete hyperbolic structure on $M$ with ends $E_1,...E_N$ admitting exactly these end invariants.

My question is, are these invariants localized in the ends ?

More precisely, given $f$ a pseudo-Anosov diffeomorphism of the surface $S_1$ and its associated lamination when $f$ is iterated in positive time. Then, one can consider the hyperbolic structure given by the ending lamination theorem associated to this lamination for the surface $S_1$ and with whatever end invariants you want for the rest of the surfaces ($S_2...S_N$).

Does the resulting hyperbolic structure satisfies that the end associated to $S_1$ is asymptotically isometric to a $\mathbb{Z}$-cover of a hyperbolic compact 3-fold which fibers over the circle with fiber $S_1$ and monodromy $f$ ?

I know from McMullen's book (theorem 3.12)

http://www.math.harvard.edu/~ctm/papers/pup.html

that this is true for quasi-Fuchsian manifolds, and I wondered whether this result may be generalised.

Even though it is still no clear to me how to deal with the case when the end invariants are the same. In fact, still looking into the case of a quasi-fuchsian groups, it is suppose to give two different hyperbolic structures to the following pairs $(S_1,S_1)$ and $(f(S_1), f(S_1))$, which seems to contradict Bers parametrisation theorem. I should be mistaking but I can't understand why.

Thank you for your reading.

Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations or conformal structures (the end invariants) on $S_1...S_n$ one is able to find a unique complete hyperbolic structure on $M$ with ends $E_1,...E_N$ admitting exactly these end invariants.

My question is, are these invariants localized in the ends ?

More precisely, given $f$ a pseudo-Anosov diffeomorphism of the surface $S_1$ and its associated lamination when $f$ is iterated in positive time. Then, one can consider the hyperbolic structure given by the ending lamination theorem associated to this lamination for the surface $S_1$ and with whatever end invariants you want for the rest of the surfaces ($S_2...S_N$).

Does the resulting hyperbolic structure satisfies that the end associated to $S_1$ is asymptotically isometric to a $\mathbb{Z}$-cover of a hyperbolic compact 3-fold which fibers over the circle with fiber $S_1$ and monodromy $f$ ?

I know from McMullen's book (theorem 3.12)

http://www.math.harvard.edu/~ctm/papers/pup.html

that this is true for one degenerated end quasi-Fuchsian manifolds or, more obviously, for $\mathbb{Z}$-cover of compact manifold I wondered whether this result may be generalised.

A very first question is what happens if we consider two pseudo-Anosov diffeomorphisms $(\Phi_1, \Phi_2)$ for which the laminations given by iterating these in positive time are in transversal position. So that the sequence of quasi-Fuchsian manifolds given in Bers coordinates by $(\Phi_1^n, \Phi_2^n)$ converges by Thurston's theorem.

Can I say that the end $E_1$ (resp $E_2$) of the limit representation are asymptotically isometric to the $Z$-covering of the compact manifold which fibres over the circle of monodromy $\Phi_1$ (resp $\Phi_2)$ ?

Thank you for your reading.