Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:
- $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
- there exists a geodesic lamination $\lambda$ on $S$ such that $g$ maps leaves of $\lambda$ to geodesics in $N$ and is totally geodesic on $S\setminus \lambda$,
- $g$ is $\pi_1$-injective.
In particular, $g$ is an isometry onto the image $g(S)$. On the other hand, $N$ has its own hyperbolic metric, so that for two points $x,y\in g(S)$ we may define two distances, say $d_\rho(x,y)$ and $d_N(x,y)$. I would like to prove that $\forall B > 0$ there exists $A > 0$ such that if $d_N(x,y)< B$ then $d_\rho(x,y)< A$.
I guess that this may be done using Lemma 4.4 in Minsky's paper "On rigidity, limit sets and end invariants of hyperbolic $3$-manifolds" which states:
Fix $S$ and $\varepsilon>0$. Given $B>0$ there exists $A$ such that if $g\colon (S,\rho)\to N$ is a pleated surface, $g_\ast$ is an isomorphism on $\pi_1$, and injectivity radii in $N$ are bounded below by $\varepsilon$, then the following holds: Let $\alpha\in S$ be a closed curve through $x\in S$, $\rho$-geodesic except possibly at $x$, and let $\beta$ denote the shortest curve in $N$ passing through $g(x)$ and homotopic to $g(\alpha)$. Then $$l_N(\beta)\le B\Rightarrow l_\rho(\alpha)\le A$$
This gives a uniform properness condition about inclusions of homotopy classes of loops from $S$ (identified with $g(S)$) into $N$. This sounds to me very similar to what I need, apart from the fact that what I need is a similar result which holds for inclusions of paths between points instead of loops. Do you think this is the right way? Could you help me with that? Thank you!