By Matthias Aschenbrenner, Stefan Friedl, Henry Wilton
The sector of 3-manifold topology has made nice strides ahead considering that 1982 whilst Thurston articulated his influential checklist of questions. fundamental between those is Perelman's evidence 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 targeted dice complexes, and, eventually, Agol's facts of the digital Haken Conjecture. This ebook summarizes these kind of advancements and offers an exhaustive account of the present cutting-edge 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 fascinating development of the final 20 years, will probably be a useful source for researchers within the box who desire a reference for those advancements. It additionally provides a fast moving advent to this fabric. even supposing a few familiarity with the elemental staff is suggested, little different past wisdom is thought, and the publication is available to graduate scholars. The ebook closes with an in depth checklist of open questions as a way to even be of curiosity to graduate scholars and confirmed researchers. A book of the ecu Mathematical Society (EMS). allotted in the Americas through the yankee Mathematical Society.
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Let x generate C, and let y ∈ π such that yC = Cy. Then yxy−1 = x±1 and hence y2 xy−2 = x. Thus x commutes with y2 , and since y2 commutes with y, we see that x commutes with y. Hence y commutes with g, thus y ∈ Cπ (g) = C. The class of CSA groups was introduced by Myasnikov–Remeslennikov[MyR96] as a natural (in the sense of first-order logic, universally axiomatizable) generalization of torsion-free word-hyperbolic groups. ) A group is said to be CSA (short for conjugately separated abelian) if all of its maximal abelian subgroups are malnormal.
1. Let ϕ ∈ SAut(H1 (T 2 ; Z)) and N = M(T 2 , ϕ). Then (1) ϕ is periodic ⇒ N Euclidean; (2) ϕ is Anosov ⇒ N is a Sol-manifold; and (3) ϕ is nilpotent ⇒ N is a Nil-manifold. The Nielsen–Thurston Classification Theorem says that if Σ is a compact, orientable surface with negative Euler characteristic, then there exists also trichotomy for elements in M(Σ). , there exists f : Σ → Σ which represents ϕ and a non-empty embedded 1-manifold Γ in Σ consisting of essential curves with an f invariant tubular neighborhood νΓ such that on each f -orbit of Σ \ νΓ the restriction of f is either finite order or pseudo-Anosov.
2. Let π be a group. If π decomposes non-trivially as an amalgamated free product π ∼ = A ∗C B, then π has a non-cyclic free subgroup unless [A : C] ≤ 2 and [B : C] ≤ 2. Similarly, if π decomposes non-trivially as an HNN-extension π ∼ = A∗C , then π contains a non-cyclic free subgroup unless one of the inclusions of C into A is an isomorphism. The proof of the lemma is a standard application of Bass–Serre theory [Ser77, Ser80]. Now we are ready to prove the theorem. 1. The implication (1) ⇒ (2) is obvious.