Pleated surfaces do not curl up too much

Question

Hi!

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:

1. $g$ is a path-isometry (it maps paths of finite length to paths of the same length),
2. there exists a geodesic lamination $\lambda$ on $S$ such that $g$ maps leaves of $\lambda$ to geodesics in $N$ and is totally geodesic on $S\setminus \lambda$,
3. $g$ 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!

Answer 1

This is an expansion of Misha's comment. Since $g : (S,\rho) \to N$ is a pleating map we have $g$ is $1$-Lipschitz. That is, for any $x, y \in S$ we have $d_N(g(x), g(y)) \leq d_\rho(x, y)$. In fact, if $\alpha$ is a geodesic arc in $(S,\rho)$ connecting $x$ to $y$, and if $\beta$ is a geodesic arc in $N$, connecting $g(x)$ to $g(y)$ in the relative homotopy class of $g(\alpha)$, then $\ell_N(\beta) \leq \ell_\rho(\alpha)$. Similarly, if $\alpha$ is a geodesic loop in $S$ then the geodesic representative $\beta$, of $g(\alpha)$, has $\ell_N(\beta) \leq \ell_\rho(\alpha)$.

It follows that $R = R_\rho$, the injectivity radius of $(S, \rho)$, is greater than or equal to the injectivity radius of $N$. Let $x, y \in S$ be any pair of points. Let $\alpha$ be the shortest geodesic segment in $S$ connecting $x$ to $y$. For any point $z$ of $\alpha$ let $D \subset S$ be the open ball of radius $R$ about $z$. It follows that $\alpha \cap D$ is a single arc, centered at $z$. Thus the $R/2$ neighborhood of $\alpha$ is an embedded strip with area less than the area of $(S, \rho)$. Since the area of $(S, \rho)$ is $-2\pi\chi(S)$, deduce an upper bound $A$ for the length of $\alpha$ and hence for the diameter of $S$.

So -

Reading the third to last sentence of your post, I think that you may be asking a different question from what you actually wrote in the first half. That is, instead of the distances $d_N(g(x), g(y))$ and $d_\rho(x,y)$, you are interested in upper bounds for certain geodesic arcs connecting the points.... Looking at Minsky's paper, I think that the proof with $x \neq y$ is basically the same compactness argument as his version with $x = y$.