Why is the dual tree to a measured geodesic lamination in a compact hyperbolic surface not complete?

Question

Let $M$ be a closed connected surface and $\mathcal{F}$ a minimal (every leaf is dense) measured foliation (as, for example, in Thurston's work on surfaces) on $M$. Let $\tilde{M}$ be the universal cover of $M$ and $\tilde{\mathcal{F}}$ the pullback of $\mathcal{F}$ to $\tilde{M}$.

Consider the leaf space of $\tilde{\mathcal{F}}$, $\mathcal{T}=\tilde{M}/\tilde{\mathcal{F}}$. Define the distance between two leaves of $\tilde{\mathcal{F}}$ as the minimum of the transverse measures of arcs joining the two leaves, we obtain a distance on $\mathcal{T}$ which turns $\mathcal{T}$ into a tree.

I have seen the following statement.

If the genus of $M$ is $\geq 2$, $\mathcal{T}$ is not complete for this distance.

Why is this true?

I also saw a notion "dual tree", which also shows up from time to time when I tried to search for related topics, from the book Hyperbolic manifolds and Discrete Groups (for geodesic laminations there). Is this indeed the concept that I am looking for?

Thanks in advance!

Answer 1

Here's a rough description of why $\mathcal T$ is incomplete.

Start at some point $x_0$ of $\tilde M$. Let $\ell_0$ be a lonnnng leaf segment of $\tilde{\mathcal F}$ with one endpoint on $x_0$, and let $y_0$ be the opposite endpoint. Let $\tau_0$ be a transverse arc of transverse measure $2^{-0}=1$ with one endpoint at $y_0$. Let $x_1$ be the opposite endpoint. Continue by induction: assuming $x_n$ has been defined, let $\ell_n$ be a lonnnng leaf segment with one endpoint on $x_n$, let $y_n$ be the opposite endpoint, let $\tau_n$ be a transverse arc of transverse measure $2^{-n}$, and let $x_{n+1}$ be the opposite endpoint. You get an infinite path of the form $$\ell_0 \tau_0 \ell_1 \tau_1 \ell_2 \tau_2 \cdots $$ The total transverse measure of the initial segment $\ell_0 \tau_0 \cdots \ell_n \tau_n$ that connects $x_0$ to $x_n$ is equal to $2-2^{-n}$, and this converges to $2$. By careful choice of the $\tau$'s in the induction step you can guarantee that this initial segment is "quasitransverse", implying that $2-2^{-n}$ is the minimum of the transverse measure of any arc from $x_0$ to $x_n$. So, in the dual tree $\mathcal T$, you get an isometric embedding $[0,2) \mapsto \mathcal T$. But, this embedding has no point to converge to as the parameter in $[0,2)$ approaches $2$; in fact, one can check that this infinite path escapes all compact subsets of $\tilde M$. So $\mathcal T$ is incomplete.

By the way, another place I know, where the dual tree is explained very clearly, is in Otal's book "Le théorème d'hyperbolisation pour les variétés fibrées de dimension 3", in the chapter on Skora's theorem.