By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead considering the fact that 1982 while Thurston articulated his influential checklist of questions. fundamental between those is Perelman's facts of the Geometrization Conjecture, yet different highlights contain the Tameness Theorem of Agol and Calegari-Gabai, the skin Subgroup Theorem of Kahn-Markovic, the paintings of clever and others on targeted dice complexes, and, eventually, Agol's facts of the digital Haken Conjecture. This publication summarizes these types of advancements and offers an exhaustive account of the present cutting-edge of 3-manifold topology, particularly concentrating on the results for primary teams of 3-manifolds. because the first ebook on 3-manifold topology that comes with the interesting development of the final twenty years, it is going to be a useful source for researchers within the box who desire a reference for those advancements. It additionally provides a fast moving creation to this fabric. even if a few familiarity with the elemental workforce is suggested, little different prior wisdom is thought, and the publication is on the market to graduate scholars. The e-book closes with an intensive checklist of open questions so that it will even be of curiosity to graduate scholars and proven researchers. A booklet of the ecu Mathematical Society (EMS). disbursed in the Americas by means of the yankee Mathematical Society.
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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).
We denote by π1orb (B ) the orbifold fundamental group of B (see [BMP03]), and following [JS79, p. 23] we refer to (p∗ )−1 (π1orb (B )) as the canonical subgroup of π1 (M). ) If f is a regular Seifert fiber of the Seifert fibration, then we refer to the subgroup of π1 (M) generated by f as the Seifert fiber subgroup of π1 (M). Recall that if M is non-spherical, then the Seifert fiber subgroup of π1 (M) is infinite cyclic and normal. ) Remark. The definition of the canonical subgroup and of the Seifert fiber subgroup depend on the Seifert fiber structure.
8. Let N be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary. (1) Let S be a boundary component of N. If the JSJ-component of N which contains S is hyperbolic, then π1 (S) is a malnormal subgroup of π1 (N). (2) Let T be a JSJ-torus of N. If both of the JSJ-components of N abutting T are hyperbolic, then π1 (T ) is a malnormal subgroup of π1 (N). The first statement was proved by de la Harpe–Weber [dlHW11, Theorem 3] and can be viewed as a strengthening of the previous theorem.