Surface groups in some surgered manifolds
For an orientable hyperbolic 3-manifold \(M\), there is a discrete faithful torsion-free image of a representation \(\rho: \pi_1(M) \to PSL_2(C)\) (the representation can be lifted to \(\pi_1(M) \to SL_2(C)\) by a result of Thurston). An immersed \(\pi_1\)-injective surface \(g: S \to M\) determines a subgroup \(\rho \circ \pi_1(g) (\pi_1(S))\) of \(PSL_2(C)\). If the subgroup \(\rho \circ \pi_1(g) (\pi_1(S))\) is conjugate into \(PSL_2(R)\), then \(S\) is a Fuchsian (totally geodesic) surface in \(M\); \(S\) is quasi-Fuchsian if the limit set of a lift of \(S\) to the universal cover is a topological circle. (1) Marsden proved that if \(M\) is a hyperbolic 3-manifold with boundary and contains a closed \(\pi_1\)-injective quasi-Fuchsian surface, then this surface will remain \(\pi_1\)-injective after sufficiently large surgeries; (2) Cooper and Long proved that if \(M\) is a finite-volume hyperbolic 3-manifold with a single cusp, then all but finitely many surgeries contain a surface group; (3) The paper under review is to characterize ``the large enough'' for the surface to remain \(\pi_1\)-injective, and to explicitly compute the 13-Dehn-surgery for the figure-eight knot complement which does not preserve the \(\pi_1\)-injective property. Note that it is well known that every manifold obtained by Dehn surgery along the figure-eight knot or the Whitehead link has a hyperbolic structure except for the six surgeries by Thurston, Neumann and Reid's results. The main theorem (Theorem 1.1 = Theorem 3.7) of the paper under review states that if any \(p/q\)-surgery curves on the boundary of a horoball neighborhood of the cusps disjoint from the projection of convex hull have representatives which are geodesic and have Euclidean length \(> 2\pi\), then any \(\pi_1\)-injective quasi-Fuchsian surface \(S\) in a complete hyperbolic 3-manifold of finite volume remains \(\pi_1\)-injective after surgery. There are two ingredients in the proof: (i) the \(2\pi\)-theorem of Thurston and Gromov a proof of which can be found in Bleiler and Hodgson's paper, (ii) for a complete Riemannian manifold \(M\) with all sectional curvature \(\leq 0\) and \(\pi_1(M) = 0\), then there are no closed geodesics. The paper lists the characterization for a closed immersed totally geodesic surface in the figure-eight knot complement through the arithmetic manifold property, and determines the size of the horoball neighborhood of the cusp. Combining this with the main theorem, the author finds out the exactly thirteen Dehn surgeries for which the \(\pi_1\)-injective surface does not remain \(\pi_1\)-injective after surgery. Similar techniques are applied to the Whitehead link and twisted knots in the last section.
scientific article; zbMATH DE number 1546936