geometrically infinite ends of hyperbolic 3 manifolds Let $M$ be a hyperbolic 3-manifold with finitely generated fundamental group. Assume $E$ is a geometrically infinite end (not of geometrically finite type, i.e. the convex core can not be separated of the end for a small neighborhood of $E$).  Then there exists a sequence of closed geodesics approaching $E$. Call this family $\mathcal{F}$.
Question: Is it true that there are closed geodesics (in the above family) that are arbitrarily far apart? To be more precise: for a closed geodesic $\gamma$ define $l_\gamma$ as the minimum distance from $\gamma$ to a geodesic in $\mathcal{F}$. Is it true that $\{l_\gamma\}_{\gamma\in\mathcal{F}}$ is unbounded?
If the answer is no, do people know hypothesis that could imply such a statement?.  
 A: You seem to be asking two different questions:


*

*The first question appears to be: "Given a family of closed geodesics ${\mathcal F}$ which exits an end $E$, is it true that there are members of ${\mathcal F}$ which are arbitrarily far apart?" 


This is true and follows directly from the definition :
A sequence of closed geodesics exits an end $E$ if for every compact $K\subset E$ all but finitely many members of the sequence are disjoint from $K$.
Now use metric neighborhoods of radius $i$ of your $\gamma$ as the compacts $K$: This yields geodesics $\gamma_i\in {\mathcal F}$ such that 
$d(\gamma, \gamma_i)\ge i$. 


*You seem to be also asking:  


"For a closed geodesic $\gamma$ define $l_\gamma$  as the minimum distance from $\gamma$  to a geodesic in ${\mathcal F}$ (I assume, different from $\gamma$). Is it true that $\{l_\gamma: \gamma\in {\mathcal F}\}$  is unbounded?"
The answer to this question is negative. An example is given by $M$ which is an infinite cyclic cover of a manifold $N$ fibered over the circle. Let $g: M\to M$ denote the generator  of the deck-group of the covering $M\to N$. Pick a geodesic closed $\gamma$ in $M$ and set ${\mathcal F}=\{\gamma_i= g^i(\gamma): i\in {\mathbb N}\}$. Then it is immediate  that for  $\gamma_i\in {\mathcal F}$ the number $l_{\gamma_i}$ is independent of $i$ and, hence, your set is bounded.  
Of course, it is entirely possible that you meant to ask yet another question (I can think of a couple question along the lines of the above), in which case you should think carefully what your question really is.  
