By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead considering that 1982 whilst Thurston articulated his influential record of questions. basic between those is Perelman's evidence 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 certain dice complexes, and, eventually, Agol's evidence of the digital Haken Conjecture. This e-book summarizes a lot of these advancements and offers an exhaustive account of the present cutting-edge of 3-manifold topology, specially targeting the implications for primary teams of 3-manifolds. because the first booklet on 3-manifold topology that includes the interesting growth of the final twenty years, it will likely be a useful source for researchers within the box who desire a reference for those advancements. It additionally provides a fast paced advent to this fabric. even though a few familiarity with the basic team is suggested, little different past wisdom is believed, and the e-book is out there to graduate scholars. The e-book closes with an intensive record of open questions so that it will even be of curiosity to graduate scholars and tested researchers. A book of the ecu Mathematical Society (EMS). disbursed in the Americas by means of the yank Mathematical Society.
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Additional info for 3-Manifold Groups
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.
Any element of A has a non-cyclic centralizer in π1 (N). 1 that N is a Seifert fibered manifold. The case of a Seifert fibered manifold then follows from an elementary argument. Remark. An infinite group π is said to be presentable by a product if there is a morphism ϕ : Γ1 × Γ2 → π onto a finite-index subgroup of π such that for i = 1, 2 the groups ϕ(Γi ) are infinite. 4], building on work of Kotschick– L¨oh [KoL09, KoL13] showed that a 3-manifold is Seifert fibered if and only if its fundamental group is presentable by a product.
We refer to [Thu97, Sco83a, KLs14] for details. The spherical and the hyperbolic manifolds in this sense are precisely the type of manifolds we introduced in the previous section. A 3-manifold is called geometric if it is an X-manifold for some geometry X. 1] a geometric 3-manifold that is either spherical, Nil or SL(2, R) is in fact orientable. No other geometries have non-orientable examples. The following theorem summarizes the relationship between Seifert fibered manifolds and geometric 3-manifolds.
3-Manifold Groups by Matthias Aschenbrenner, Stefan Friedl, Henry Wilton