Download e-book for iPad: Algebra IV: infinite groups, linear groups by A. I. Kostrikin, I. R. Shafarevich

By A. I. Kostrikin, I. R. Shafarevich

ISBN-10: 0387533729

ISBN-13: 9780387533728

Crew concept is among the so much primary branches of arithmetic. This quantity of the Encyclopaedia is dedicated to 2 very important matters inside of staff idea. the 1st a part of the e-book is anxious with endless teams. The authors take care of combinatorial workforce concept, loose buildings via staff activities on timber, algorithmic difficulties, periodic teams and the Burnside challenge, and the constitution concept for Abelian, soluble and nilpotent teams. they've got integrated the very most recent advancements; notwithstanding, the fabric is offered to readers acquainted with the elemental options of algebra. the second one half treats the speculation of linear teams. it's a really encyclopaedic survey written for non-specialists. the themes lined contain the classical teams, algebraic teams, topological equipment, conjugacy theorems, and finite linear teams. This publication can be very worthwhile to all mathematicians, physicists and different scientists together with graduate scholars who use staff thought of their paintings.

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Extra info for Algebra IV: infinite groups, linear groups

Example text

8, α intersects β in exactly one point x. It cannot be that the axes of the hyperbolic isometries corresponding to α and β share exactly one endpoint at ∂H2 , because this would violate discreteness of the action of π1 (S) on H2 ; indeed, in this case the commutator of these isometries is parabolic and the conjugates of this parabolic isometry by either of the original hyperbolic isometries have arbitrarily small translation length. Further, these axes cannot share two endpoints on ∂H2 , for then the corresponding hyperbolic isometries would have the same axis, and so they would have to have a common power φ (otherwise the action of π1 (S) on this axis would be nondiscrete).

Since ∂D embeds under the covering map, φ(∂D) ∩ ∂D is either empty or all of ∂D (in the case that φ is the identity). By the Jordan curve theorem, we then see that either φ(D) or φ−1 (D) must be contained in D. Now, by the Brouwer fixed point theorem, φ has a fixed point, which is a contradiction, unless φ is the identity. ✷ We give two proofs of the bigon criterion. One proof uses hyperbolic geometry and one proof uses only topology. We give both proofs since each of the techniques will be important later in this book.

In particular it makes sense to write ˆi(a, b) for a and b the free homotopy classes (or homology classes) of closed curves α and β. The most naive way to count intersections between homotopy classes of closed curves is to simply count the minimal number of unsigned intersections. This idea is encoded in the concept of geometric intersection number. The geometric intersection number between free homotopy classes a and b of simple closed curves in a surface S is defined to be the minimal number 30 CHAPTER 1 of intersection points between a representative curve in the class a and a representative curve in the class b: i(a, b) = min{|α ∩ β| : α ∈ a, β ∈ b}.

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Algebra IV: infinite groups, linear groups by A. I. Kostrikin, I. R. Shafarevich

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