By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

The sphere of 3-manifold topology has made nice strides ahead on the grounds that 1982 while Thurston articulated his influential checklist 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 targeted dice complexes, and, ultimately, Agol's evidence of the digital Haken Conjecture. This booklet summarizes these kinds of advancements and offers an exhaustive account of the present state-of-the-art of 3-manifold topology, specifically targeting the implications for primary teams of 3-manifolds. because the first ebook on 3-manifold topology that includes the intriguing development of the final 20 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 moving creation to this fabric. even supposing a few familiarity with the elemental crew is usually recommended, little different prior wisdom is believed, and the e-book is on the market to graduate scholars. The publication closes with an in depth record of open questions in an effort to even be of curiosity to graduate scholars and demonstrated researchers. A booklet of the eu Mathematical Society (EMS). dispensed in the Americas through the yankee Mathematical Society.

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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. , 3-manifolds that are compact, orientable, and irreducible, with empty or toroidal boundary.

More precisely, we will describe all fundamental groups of compact 3-manifolds that are CA and CSA. A group is said to be CA (short for centralizer abelian) if the centralizer of any non-identity element is abelian. Equivalently, a group is CA if and only if the intersection of any two distinct maximal abelian subgroups is trivial, if and only if ‘commuting’ is an equivalence relation on the set of non-identity elements. For this reason, CA groups are also sometimes called ‘commutative transitive groups’ (or CT groups, for short).

So let T be a surface in T . 2 takes care of all cases, except the following: T corresponds to a toroidal component of S such that S := p(S ) is a Klein bottle. We turn our attention to this special case. By definition S cobounds a copy X of K 2 × I in N. By the above N is not a twisted double of K 2 × I. It follows that S also bounds a component M of N cut along S that is not homeomorphic to K 2 × I. We denote by M a component of p−1 (M) that bounds S . Furthermore we pick an identification of K 2 × I with the twisted Klein bottle cobounded by S.