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

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.

Show description

Read or Download 3-Manifold Groups PDF

Similar group theory books


This e-book is a finished advent to the speculation of good commutator size, a massive subfield of quantitative topology, with titanic connections to 2-manifolds, dynamics, geometric workforce conception, bounded cohomology, symplectic topology, and lots of different matters. We use optimistic equipment every time attainable, and concentrate on basic and specific examples.

Geometry and Cohomology in Group Theory

This quantity displays the fruitful connections among workforce idea and topology. It includes articles on cohomology, illustration idea, geometric and combinatorial workforce thought. a number of the world's top recognized figures during this very energetic region of arithmetic have made contributions, together with huge articles from Ol'shanskii, Mikhajlovskii, Carlson, Benson, Linnell, Wilson and Grigorchuk that would be invaluable reference works for a few future years.

Rings, modules, and algebras in stable homotopy theory

This booklet introduces a brand new point-set point method of reliable homotopy concept that has already had many purposes and can provide to have a long-lasting impression at the topic. Given the sector spectrum $S$, the authors build an associative, commutative, and unital spoil product in a whole and cocomplete class of ""$S$-modules"" whose derived type is corresponding to the classical good homotopy classification.

Additional resources for 3-Manifold Groups

Example text

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.

Download PDF sample

Rated 4.02 of 5 – based on 18 votes