Two-parabolic-generator subgroups of hyperbolic \(3\)-manifold groups (Q6614434)

The authors give a detailed account of Agol's theorem and proof concerning two-meridional-generator subgroups of hyperbolic 2-bridge link groups (as presented in Agol's talk at the Budapest Bolyai conference 2001).\N\NBy a result of \textit{C. C. Adams} [Commun. Anal. Geom. 4, No. 2, 181--206 (1996; Zbl 0863.57006)], the fundamental group of a finite volume hyperbolic 3-manifold is generated by two parabolic elements if and only if the 3-manifold is homeomorphic to the complement of a 2-bridge non-torus link; also, the pair consists of meridians, and each hyperbolic 2-bridge link group admits only finitely many distinct parabolic generating pairs up to equivalence. In a talk given at the Budapest Bolyai conference in July 2001 (``The classification of non-free 2-parabolic generator Kleinian groups''), Agol generalizes and refines these results: Given a hyperbolic 2-bridge link, any non-commuting meridian pair in the link group which is not equivalent to the upper nor lower meridian pair generates a free Kleinian group which is geometrically finite.\N\N``The main purpose of the present paper is to give a detailed account of Agol's beautiful proof of his theorem. A key ingredient of the proof is non-positively curved cubed decompositions of alternating link exteriors in which the checkerboard surfaces are hyperplanes.''