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.
- 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)?
- 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?
- 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 :-(