Surface groups in some surgered manifolds
type: Item
instance of: scholarly article
title: Surface groups in some surgered manifolds
zbMATH Open document ID: 0961.57016
DOI: 10.1016/S0040-9383(99)00062-2
author: Anneke Bart
published in: Topology
publication date: 2001-03-19T00:00:00Z
review text: 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 Thurst…
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57R65
Mathematics Subject Classification ID: 57N35
zbMATH DE Number: 1546936
zbMATH Keywords: \(\pi_1\)-injective surface
zbMATH Keywords: quasi-Fuchsian surface
zbMATH Keywords: Dehn surgery
zbMATH Keywords: arithmetic manifold
zbMATH Keywords: totally geodesic surface
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1546936