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As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:

Let $M$ be a compact hyperbolic $3$-manifold. Then for any $x\in M$ and for any direction $v\in T_xM$ one may find an immersed surface $S$ in $M$ which is: arbitrarily $\varepsilon$-quasi-Fuchsian, passes arbitrarily closes to $x$ and with normal arbitrarily close to $v$.

First question: do I understand correctly?

Second question: where I can find good reference for such kind of results? Do there exist surveys which are more accessible than the original paper to a broad audience?

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I think that a good reference could be the following:

Proposition ([1, Proposition 4.1 (or 5.1 in the arXiv version)]): Let $M$ be a closed hyperbolic 3-manifold, and regard $\pi_1(M)$ as acting on $\mathbb{H}^3 \simeq \widetilde{M}^3$. For each great circle $C \subset \partial \mathbb{H}^3$ there is a sequence of immersions $(f_i \colon F_i \to M)_i$, where each $F_i$ is a surface and $f_*(\pi_1(F_i))$ is a quasi-Fuchsian subgroup in $\pi_1(M)$, such that $\partial \widetilde{f}_i (\widetilde{F}_i) \subset \mathbb{H}^3$ pointwise converges to $C$.

The "pointwise convergence" can be misleading; the proof of Corollary 4.2 (that you can find right after in the paper) clarifies this notion.

The authors say that this is implied by [2, Theorem 1.1], which should be the same as [3, Theorem 1.1]. To me, this implication is not crystal clear; I think that an expert will be able to clarify this.

Once you have the Proposition, your result should follow easily: take a point and a tangent vector in $\mathbb{H}^3$ that map to your point and vector in $M$. Consider the plane perpendicular to this vector. The sequence of surfaces given by the Proposition should satisfy your requests (to be 100% sure, I should ask which notion of $\varepsilon$-quasi-Fuchsian we are using).

[1]: Bergeron, Nicolas, and Daniel T. Wise. "A boundary criterion for cubulation." American Journal of Mathematics 134.3 (2012): 843-859.

[2]: J. Kahn and V. Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, preprint, 2009.

[3]: Kahn, Jeremy, and Vladimir Markovic. "Immersing almost geodesic surfaces in a closed hyperbolic three manifold." Annals of Mathematics (2012): 1127-1190.

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  • $\begingroup$ Yes, the Proposition in Bergeron Wise is more or less equivalent to my statement. The problem is that the proposition starts with "for each gret circle" while Theorem 1.1. in Kahn Markivich uses "there is". So strictly speaking there are wrong quantifiers. However, it seems to me that in the Kahn Markovich paper, they uses equidistribution properties of the geodesic flows, so I understand that the "there is" statement can be upgraded to a "for all" statement. I don't read that so explicitly in the paper. I wonder if there is some survey where this has been stated (and proved) explicitly. $\endgroup$
    – user126154
    Nov 16, 2022 at 11:19

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