Immersed quasi-Fuchsian surfaces surviving Dehn fillings

Question

In papers like, Cooper - Long - Some surface subgroups survive surgery or Li - Immersed essential surfaces in hyperbolic 3-manifolds the game is to find some quasi-Fuchsian immersed surface $Q \looparrowright M$ where $M$ is a hyperbolic 3-manifold with toroidal cusps and then to Dehn fill $M$ along some (multi) slope $\gamma$ that is sufficiently far from the immersed boundary slope(s) of $Q$. In the Dehn-filled quotient $M(\gamma)$ something clever is done to the image of $Q$ to produce a closed essential (= immersed + $\pi_1$-injective) surface $F\looparrowright M(\gamma)$. Doing so was justifiably a source of pride, this was pre Kahn-Markovich.

I am curious about how the techniques of these papers, or others, could be used to answer a completely different question. Here's what I want to do: I have an atoroidal 3-manifold with torus boundary components (i.e. its interior is hyperbolizable) and a fixed immersed boundary slope $\gamma$, denote by $f:M \hookrightarrow M(\gamma)$ the inclusion into the Dehn filling. Can I find some essential surface with boundary $i:(S,\partial S) \looparrowright (M,\partial M)$ such that $f \circ i$ is still $\pi_1$-injective? I know that this composition is a map from a surface with boundary to a closed 3-manifold (i.e. with empty boundary), which is a weird thing to do for a 3-manifold person, but I'm actually more interested by the group theory.

The literature is technically difficult for me and I am unclear about a few things.

1. Both linked papers have a Theorem 1.2 that seems to say that if $M$ is a hyperbolic 3-manifold and $\alpha$ is the immersed boundary slope of some essential $Q\looparrowright M$ then if a filling slope $\gamma$ is "far" enough from $\alpha$ we can cook up an essential closed surface in $M(\gamma)$. A hyperbolic knot complement has a Dehn filling to the 3-sphere, does this mean that such 3-manifolds only admit finitely many immersed boundary slopes, since they all need to be close to the meridian (the slope that fills to give a sphere)?
2. There is a result due to Baker, as well as many generalizations, that in some cases infinitely many immersed boundary slopes from essential surfaces may occur. This phenomenon seems opposite to knot complements. Is there a characterization of the hyperbolic 3-manifolds that admit an infinity of immersed boundary slopes?
3. If the construction given in 1. above goes through (i.e. the immersed boundary slope is far from the filling slope) is it know whether $Q$ (which the authors stop caring about) also $\pi_1$-injects into $M(\gamma)$?

Again, just to be clear because it's weird, I have a fixed Dehn filling and I want to find essential surfaces with "far" immersed boundary slopes. Any help, even pointing out that I've got it all wrong, would be greatly appreciated. Might as well assume that $M$ has one boundary component.

EDIT: Please ignore 1. I misread Theorem 1.2 in both papers. The quasi-Fuchsian surface $Q$ is embedded, the mystery closed surface that is constructed is immersed. This is embarrassing: I got it all wrong :-(

Answer 1

Notice that Hatcher proves that three-manifolds with a single torus boundary have only finitely many embedded boundary slopes. So I assume that your question one is asking about "immersed boundary slopes": the slopes occurring as boundaries of essential immersed surfaces. You ask if knot complements have only finitely many such. The answer is "no" for two-bridge knots. See the paper "Virtually embedded boundary slopes" by Joseph Maher (http://arxiv.org/abs/math/9901041). (NB - Maher uses boundary slope as a synonym for "slope".)

Regarding question two: I don't have a reference or proof at hand, but I'll boldly guess that all once-cusped hyperbolic three-manifolds have infinitely many immersed boundary slopes.

Regarding question three: Again, I think your order of quantifiers is strange here. It doesn't matter how "far" the immersed boundary slope is from the filling slope, if the filling slope is, say, the meridian slope for the knot complement.

Answer 2

For question 2., one can prove that for a 1-cusped orientable hyperbolic 3-manifold $M$, all but finitely many slopes bound an immersed essential surface. This follows by combining the hyperbolic Dehn filling theorem (in fact, the $2\pi$ or 6 theorems suffice) and Kahn-Markovic's theorem. If the slope $\alpha$ has length $> 6$ in an embedded cusp, then the core of the Dehn filling $\gamma$ will be infinite order and the filling $M(\alpha)$ will be hyperbolic. By Kahn-Markovic, there is an immersed $\pi_1-$injective closed orientable surface $f:\Sigma \to M(\alpha)$. Assume that $f\pitchfork\gamma$, then pass to the covering space $ \pi: N\to M(\alpha)$ so that $f$ lifts to a map $\tilde{f}:\Sigma \to N$ which is a homotopy equivalence and an embedding. Then $\tilde{f} \cap \pi^{-1}(\gamma)$ consists of finitely many points. If $\tilde{f}\cap N-\pi^{-1}(\gamma)$ is not incompressible in $N-\pi^{-1}(\gamma)$, then one may perform compressions via an isotopy in $N$ (since $N$ is irreducible and $\tilde{f}$ incompressible) reducing the number of intersections until $\tilde{f}\cap N-\pi^{-1}(\gamma)$ is incompressible in the complement of $N-\pi^{-1}(\gamma)$. Then this surface will push down to $M$ to give a surface with immersed boundary slope $\alpha$. One might object that the surface might be homotoped in $M(\alpha)$ to completely miss $\gamma$. But one may use properties of the Kahn-Markovic surfaces and the fact that $\gamma$ is essential in $M(\alpha)$ to show that this doesn't happen if the surface is close enough to being geodesic (via an argument of Bergeron-Wise).

The more recent paper of Cooper-Futer proves the existence of closed quasi-fuchsian surfaces in $M$ using the Kahn-Markovic surfaces of Dehn fillings as a starting point (so similar to this argument).

Answer 3

Baker and Cooper arXiv:math/0507004 MR2417445 (2009i:57035) A combination theorem for convex hyperbolic manifolds, with applications to surfaces in 3-manifolds.
J. Topol. 1 (2008), no. 3, 603–642.

In the final sections of the paper, ... (3) showing that if the interior of a compact manifold with torus boundary admits a hyperbolic metric, then every slope on the boundary torus is a multiple immersed boundary slope (MIBS).

This means there is an essential immersed surface with two boundary components each of which is a power of $\gamma$.