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

By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton

The sphere of 3-manifold topology has made nice strides ahead seeing that 1982 whilst Thurston articulated his influential checklist of questions. fundamental between those is Perelman's facts 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 specified dice complexes, and, eventually, Agol's evidence of the digital Haken Conjecture. This booklet summarizes a majority of these advancements and offers an exhaustive account of the present state-of-the-art of 3-manifold topology, in particular concentrating on the implications for basic teams of 3-manifolds. because the first ebook on 3-manifold topology that includes the interesting development 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 provides a fast paced advent to this fabric. even supposing a few familiarity with the basic staff is usually recommended, little different prior wisdom is believed, and the e-book is out there to graduate scholars. The ebook closes with an intensive record of open questions so as to even be of curiosity to graduate scholars and tested researchers. A e-book of the ecu Mathematical Society (EMS). allotted in the Americas via the yankee Mathematical Society.

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No element of π1 (N)\{1} is divisible by infinitely many n. Remark. 3], [Shn75, p. 327] and [Ja75, p. 328]; see also [Wan69] and [Swp73, Feu76a, Feu76b, Feu76c]. 1, the fundamental group of a non-spherical Seifert fibered manifold has a normal infinite cyclic subgroup, namely its Seifert fiber subgroup. 1 shows that the converse holds. 5. Let N be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary. If π1 (N) has a normal infinite cyclic subgroup, then N is Seifert fibered.

In many cases, however, it is difficult to obtain topological information about N by just applying group-theoretical methods to π1 (N). 5 Centralizers 35 given a closed 3-manifold N, the minimal number r(N) of generators of π1 (N) is a lower bound on the Heegaard genus g(N) of N. It has been a long standing question of Waldhausen to determine for which 3-manifolds the equality r(N) = g(N) holds. (See [Hak70, p. ) The case r(N) = 0 is equivalent to the Poincar´e conjecture. It has been known for a while that r(N) = g(N) for graph manifolds [BoZ83, BoZ84, Zie88, Mon89, Wei03, ScW07, Won11], and evidence for the inequality for some hyperbolic 3-manifolds was given in [AN12, Theorem 2].

The proof of the above theorem shows that every compact 3-manifold with nilpotent fundamental group is either spherical, Euclidean, or a Nil-manifold. Using the discussion of these geometries in [Sco83a] one can then determine the list of nilpotent groups which can appear as fundamental groups of compact 3-manifolds. 1] in general. We conclude this section with a discussion of 3-manifolds with abelian fundamental groups. In order to do this we first recall the definition of lens spaces. Given coprime natural numbers p and q we denote by L(p, q) the corresponding lens space, defined as L(p, q) := S3 /Z p = {(z1 , z2 ) ∈ C2 | |z1 |2 + |z2 |2 = 1}/Z p , where k ∈ Z p acts on S3 by (z1 , z2 ) → (z1 e2πik/p , z2 e2πikq/p ).

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