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Download e-book for kindle: 3-Manifold Groups by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

ISBN-10: 3037191546

ISBN-13: 9783037191545

The sphere of 3-manifold topology has made nice strides ahead considering the fact that 1982 whilst Thurston articulated his influential checklist of questions. fundamental between those is Perelman's facts of the Geometrization Conjecture, yet different highlights comprise the Tameness Theorem of Agol and Calegari-Gabai, the skin Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on designated dice complexes, and, eventually, Agol's evidence of the digital Haken Conjecture. This publication summarizes some of these advancements and offers an exhaustive account of the present cutting-edge of 3-manifold topology, particularly targeting the implications for basic teams of 3-manifolds. because the first e-book on 3-manifold topology that includes the intriguing growth of the final 20 years, it will likely be a useful source for researchers within the box who desire a reference for those advancements. It additionally offers a fast moving advent to this fabric. even supposing a few familiarity with the basic team is suggested, little different earlier wisdom is believed, and the ebook is on the market to graduate scholars. The e-book closes with an intensive record of open questions with a view to even be of curiosity to graduate scholars and proven researchers. A booklet of the ecu Mathematical Society (EMS). allotted in the Americas via the yank Mathematical Society.

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The geometry of a mapping torus can be studied in terms of the monodromy. We start out with the ‘baby case’ that Σ is the torus T 2 = S1 × S1 . 5] for details. Given ϕ ∈ SAut(H1 (T 2 ; Z)) it follows from an elementary linear algebraic argument that one of the following occurs: (1) ϕ n = id for some n ∈ {1, 2, 4, 6}; or (2) ϕ has two distinct real eigenvalues; or (3) ϕ is non-diagonalizable but has eigenvalue ±1. Accordingly we say that ϕ is periodic, or nilpotent, or Anosov. We then have the following geometrization theorem for torus bundles.

Hyperbolic’). 6. He introduced the notion of a geometry of a 3-manifold and of a geometric 3-manifold. We give a quick summary of the definitions and the most relevant results, and refer to the expository papers by Scott [Sco83a] and Bonahon [Bon02] as well as to Thurston’s book [Thu97] for proofs and further references. A 3-dimensional geometry is a smooth, simply connected 3-manifold X equipped with a smooth, transitive action of a Lie group G by diffeomorphisms on X, with compact point stabilizers.

13] any simply connected spherical 3-manifold is isometric to S3 . 4 (Poincar´e Conjecture). Each closed, simply connected 3-manifold is homeomorphic to S3 . , [Whd34, Whd35a, Kos58, Sta66, Vo96, Sz08, Vo14] for details. Now we turn to atoroidal 3-manifolds with infinite fundamental groups. 5 (Hyperbolization Theorem). Let N be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary. Suppose that N is atoroidal and not homeomorphic to S1 × D2 , T 2 × I, or K 2 × I. If π1 (N) is infinite, then N is hyperbolic.

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3-Manifold Groups by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

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