Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 3: Manifolds that Fiber over the Circle, タイヒミュラー理論と幾何学、トポロジー、および 動力学への応用, 第3巻:円上のファイバーの多様体, 9781943863013, 978-1-943863-01-3
学術書籍 理工学 Teichmüller Theory: タイヒミュラー理論
Teichmüller Theory and Applications to Geometry, Topology, and Dynamics, Volume 3
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Description
This book shows that a 3-manifold that fibers over the circle has a hyperbolic structure if and only if the holonomy of the fibering is pseudo-Anosov. Proving this result requires a lot of background.
The first chapter concerns hyperbolic geometry and Kleinian groups. Topics include Jorgensen's inequality, the Margulis lemma, algebraic and especially geometric limits with both the Chabauty and the Thurston-Gromov approaches, the Klein-Maskit combination theorems, the Poincare polyhedron theorem, and geometrically finite Kleinian groups.
The second chapter covers rigidity theorems: Ahlfors, McMullen, and Mostow. Verifying that quasi-Fuchsian groups satisfy the hypothesis of the McMullen rigidity theorem is a long and beautiful trip through laminations and pleated surfaces.
The third chapter proves the main result, visiting on the way the compactness of Bers slices, R-trees, Chiswell functions, Hatcher's construction and Skora's theorem, and the Otal compactness theorem.
The appendices take up 200 pages. Topics include the Nullstellensatz and Selberg's lemma, the relation between the ideal class group and the ends of Bianchi manifolds, the Haudorff dimension (1) of the space of simple geodesics on a hyperbolic surface, period coordinates, the ergodic theorem and Hopf's argument, the volume of strata of quadratic differentials, a minimal measured foliation that is not ergodic, and the Thurston norm and its relation to hyperbolic structures.
You will find a brief introduction to Chapters 11-13 below, along with a sampling of illustrations.
