
David Fisher (ed.)
et al.
Published online:
19 May 2022
Published in print:
07 February 2022
Online ISBN:
9780226804163
Print ISBN:
9780226804026
Contents
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1 Introduction 1 Introduction
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2 Kleinian groups 2 Kleinian groups
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2.1 Lattices. 2.1 Lattices.
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2.2 Finitely Generated Kleinian Groups. 2.2 Finitely Generated Kleinian Groups.
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2.3 Examples of Geometrically Finite Groups. 2.3 Examples of Geometrically Finite Groups.
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2.3.1 Schottky groups 2.3.1 Schottky groups
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2.3.2 Fuchsian groups and deformations: Quasi-Fuchsian groups 2.3.2 Fuchsian groups and deformations: Quasi-Fuchsian groups
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2.3.3 Rigid acylindrical groups and their deformations 2.3.3 Rigid acylindrical groups and their deformations
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2.4 Thurston’s Geometrization Theorem. 2.4 Thurston’s Geometrization Theorem.
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2.5 Density of Geometrically Finite Groups. 2.5 Density of Geometrically Finite Groups.
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2.6 Examples of Geometrically Infinite Groups. 2.6 Examples of Geometrically Infinite Groups.
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3 Mixing and classification of N-orbit closures 3 Mixing and classification of N-orbit closures
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3.1 Patterson-Sullivan Density. 3.1 Patterson-Sullivan Density.
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3.2 Mixing of the Bms Measure. 3.2 Mixing of the Bms Measure.
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3.3 Essential Unique Ergodicity of the Br Measure. 3.3 Essential Unique Ergodicity of the Br Measure.
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3.4 Closures of N-Orbits. 3.4 Closures of N-Orbits.
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4 Almost all results on orbit closures 4 Almost all results on orbit closures
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4.1 Geodesic Planes and Almost All H-Orbits. 4.1 Geodesic Planes and Almost All H-Orbits.
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4.2 Horocycles and Almost All U-Orbits. 4.2 Horocycles and Almost All U-Orbits.
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4.3 Orbit Closure Theorem for Lattices. 4.3 Orbit Closure Theorem for Lattices.
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4.4 Topological Obstructions to Orbit Closure Theorem. 4.4 Topological Obstructions to Orbit Closure Theorem.
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4.4.1 Fuchsian 3-manifolds 4.4.1 Fuchsian 3-manifolds
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4.4.2 Quasi-Fuchsian hyperbolic 3-manifolds 4.4.2 Quasi-Fuchsian hyperbolic 3-manifolds
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5 Unipotent blowup and renormalizations 5 Unipotent blowup and renormalizations
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5.1 Use of Unipotent Blowup in the Compact Γ\G Case. 5.1 Use of Unipotent Blowup in the Compact Γ\G Case.
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5.2 Unipotent Blowup and Renormalizations of the Return Time. 5.2 Unipotent Blowup and Renormalizations of the Return Time.
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5.3 Relative Minimal Sets and Additional Invariance. 5.3 Relative Minimal Sets and Additional Invariance.
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6 Interior frames and boundary frames 6 Interior frames and boundary frames
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6.1 Interior Frames. 6.1 Interior Frames.
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6.2 Boundary Frames. 6.2 Boundary Frames.
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7 Rigid acylindrical groups and circular slices of Λ 7 Rigid acylindrical groups and circular slices of Λ
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7.1 Boundary Frames for Rigid Acylindrical Groups. 7.1 Boundary Frames for Rigid Acylindrical Groups.
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7.2 Circular Slices of Λ. 7.2 Circular Slices of Λ.
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7.3 Closed or Dense Dichotomy for H-Orbits. 7.3 Closed or Dense Dichotomy for H-Orbits.
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7.4 Topological Rigidity of Geodesic Planes. 7.4 Topological Rigidity of Geodesic Planes.
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7.5 Classification of U-Orbit Closures. 7.5 Classification of U-Orbit Closures.
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7.6 Measure Rigidity? 7.6 Measure Rigidity?
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8 Geometrically finite acylindrical hyperbolic 3-manifolds 8 Geometrically finite acylindrical hyperbolic 3-manifolds
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8.1 Rigidity of Geodesic Planes in M*. 8.1 Rigidity of Geodesic Planes in M*.
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8.2 Closed or Dense Dichotomy for Acylindrical Groups. 8.2 Closed or Dense Dichotomy for Acylindrical Groups.
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9 Unipotent flows in higher dimensional hyperbolic manifolds 9 Unipotent flows in higher dimensional hyperbolic manifolds
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9.1 Convex Cocompact Hyperbolic Manifolds with Fuchsian Ends. 9.1 Convex Cocompact Hyperbolic Manifolds with Fuchsian Ends.
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9.2 Orbit Closure of Unipotent Flows are Relatively Homogeneous. 9.2 Orbit Closure of Unipotent Flows are Relatively Homogeneous.
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9.3 Avoidance of Singular Set. 9.3 Avoidance of Singular Set.
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9.4 Induction. 9.4 Induction.
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References References
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Chapter
14 Dynamics for Discrete Subgroups of SL2(ℂ): Dedicated to Gregory Margulis with affection and admiration
Get access
Pages
506–566
-
Published:February 2022
Cite
OXFORD ACADEMIC STYLE
Fisher, David, Dmitry Kleinbock, and Gregory Soifer (eds), 'Dynamics for Discrete Subgroups of SL2(ℂ): Dedicated to Gregory Margulis with affection and admiration', in David Fisher, Dmitry Kleinbock, and Gregory Soifer (eds), Dynamics, Geometry, Number Theory: The Impact of Margulis on Modern Mathematics (Chicago, IL , 2022; online edn, Chicago Scholarship Online, 19 May 2022), https://doi.org/10.7208/chicago/9780226804163.003.0015, accessed 24 Apr. 2025.
CHICAGO STYLE
Fisher, David, Dmitry Kleinbock, and Gregory Soifer (eds). "Dynamics for Discrete Subgroups of SL2(ℂ): Dedicated to Gregory Margulis with affection and admiration." In Dynamics, Geometry, Number Theory: The Impact of Margulis on Modern Mathematics. Edited by David Fisher, Dmitry Kleinbock, and Gregory Soifer (eds). University of Chicago Press, 2022. Chicago Scholarship Online, 2022. https://doi.org/10.7208/chicago/9780226804163.003.0015.
Abstract
Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works.
Subject
Geometry
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