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 1982 while Thurston articulated his influential checklist of questions. basic 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 detailed dice complexes, and, eventually, Agol's facts of the digital Haken Conjecture. This booklet summarizes these types of advancements and gives an exhaustive account of the present cutting-edge of 3-manifold topology, specifically concentrating on the implications for primary teams of 3-manifolds. because the first e-book on 3-manifold topology that comes with the intriguing development of the final twenty years, it will likely be a useful source for researchers within the box who want a reference for those advancements. It additionally provides a fast paced advent to this fabric. even if a few familiarity with the elemental workforce is suggested, little different earlier wisdom is thought, and the ebook is available to graduate scholars. The publication closes with an intensive checklist of open questions to be able to even be of curiosity to graduate scholars and tested researchers. A ebook of the eu Mathematical Society (EMS). allotted in the Americas by way of the yankee Mathematical Society.

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Then the geometric decomposition surface of N is given by T1 ∪ · · · ∪ Tn ∪ Kn+1 ∪ · · · ∪ Km . Conversely, if T1 ∪ · · · ∪ Tn ∪ Kn+1 ∪ · · · ∪ Km is the geometric decomposition such that T1 , . . , Tn are tori and Kn+1 , . . 9 The Geometric Decomposition Theorem 21 JSJ-tori are given by T1 ∪ · · · ∪ Tn ∪ ∂ νKn+1 ∪ · · · ∪ ∂ νKm . 3. Let N be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary such that N = S1 × D2 , N = T 2 × I, and N = K 2 × I. Let p : N → N be a finite cover and let S be the geometric decomposition surface of N.

4], yields that r(N) = g(N) for ‘sufficiently complicated’ hyperbolic 3-manifolds. ) Recently Li [Lia13] showed that there also exist hyperbolic 3manifolds with r(N) < g(N). See [Shn07] for some background. 5 Centralizers Let π be a group. The centralizer of a subset X of π is defined to be the subgroup Cπ (X) := {g ∈ π : gx = xg for all x ∈ X} of π. For x ∈ G we also write Cπ (x) := Cπ ({x}). Determining the centralizers is often one of the key steps in understanding a group. In the world of 3-manifold groups, thanks to the Geometrization Theorem, an almost complete picture emerges.

It follows easily that S and S are also isotopic. 1 as the geometric decomposition surface of N. 6 and vice versa. 1 gives us the following proposition. 2. Let N be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary with N = S1 × D2 , N = T 2 × I, and N = K 2 × I. , the geometric decomposition surface is empty. On the other hand N has one JSJ-torus, namely, if N is a torus bundle, then the fiber is the JSJ-torus, and if N is a twisted double of K 2 × I, then the JSJ-torus is given by the boundary of K 2 × I.

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

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