By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

The sphere of 3-manifold topology has made nice strides ahead seeing that 1982 while Thurston articulated his influential record of questions. fundamental between those is Perelman's evidence of the Geometrization Conjecture, yet different highlights contain the Tameness Theorem of Agol and Calegari-Gabai, the outside Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on particular dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This booklet summarizes these kind of advancements and gives an exhaustive account of the present state-of-the-art of 3-manifold topology, in particular concentrating on the results for primary teams of 3-manifolds. because the first publication on 3-manifold topology that includes the interesting 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 paced advent to this fabric. even supposing a few familiarity with the basic workforce is usually recommended, little different earlier wisdom is thought, and the publication is offered to graduate scholars. The e-book closes with an intensive record of open questions for you to even be of curiosity to graduate scholars and verified researchers. A e-book of the eu Mathematical Society (EMS). dispensed in the Americas by means of the yankee Mathematical Society.

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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. The Lie group G is called the group of isometries of X. A geometric structure on a 3-manifold N (modeled on X) is a diffeomorphism from the interior of N to X/π, where π is a discrete subgroup of G acting freely on X.

5 contains a description of the centralizers of elements of 3-manifold groups, and some applications. 1 Closed 3-manifolds and fundamental groups It is well known that closed, compact surfaces are determined by their fundamental groups; more generally, compact surfaces with possibly non-empty boundary are determined by their fundamental groups together with the number of boundary components. ) In 3manifold theory a similar, but more subtle, picture emerges. There are several ways for constructing pairs of compact, orientable, non-diffeomorphic 3-manifolds with isomorphic fundamental groups.

Proof. 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.