Two-parabolic-generator subgroups of hyperbolic \(3\)-manifold groups
type: Item
instance of: scholarly article
title: Two-parabolic-generator subgroups of hyperbolic \(3\)-manifold groups
author: Shunsuke Sakai
author: Makoto Sakuma
published in: Hiroshima Mathematical Journal
publication date: 2024-10-07T00:00:00Z
review text: 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.''
reviewed by: Bruno Zimmermann
Mathematics Subject Classification ID: 57K32
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 57K10
zbMATH DE Number: 7922230
zbMATH Keywords: alternating link
zbMATH Keywords: cubed complex
zbMATH Keywords: parabolic transformation
zbMATH Keywords: two-bridge link
MaRDI profile type: Publication
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DOI: 10.32917/H2023002
description: scientific article; zbMATH DE number 7922230