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Kleinian Groups

  • Michael Kapovich
Chapter
Part of the Modern Birkhäuser Classics book series (MBC)

Abstract

Our discussion of nilpotent groups follows [War76]. If G is a group with two subsets A, B, then [A, B,] denotes the subgroup generated by all elements of the form [a, b], a ϵ A, b ϵ B. In the special case A = B = G, the subgroup [G, G] is called the commutator subgroup of G. Suppose that Г is a group. Recall that a Central Series of Г is a descending chain of subgroups
$$ \cdots \subset \Gamma _3 \subset \Gamma _2 \subset \Gamma _1 = \Gamma ,$$
Where each Г i+1 is normal in Г i and [Г, Г i ] ⊂ Г i+1. A group is nilpotent if it admits a finite central series
$$1 = \Gamma _{n + 1} \subset \cdots \subset \Gamma _3 \subset \Gamma _2 \subset \Gamma _1 = \Gamma .$$
The smallest number n for which this property holds is called the nilpotent class of Г. If Г is nilpotent of the class n, then Г is said to be n + 1-step nilpotent. Of spectial importantance are two types of central series: (a) the lower central series Г i+1 :=[Г,Г i ] and (b) the the upper central series, where the subgroups Г i :=Zi(Г) are defined as follows. Z n (Г) is the center of Г. Inductively, Zi–1(Г) :={x ϵ Г : xZi(Г)) ϵ center (Г/Zi(Г)}}. It is elementary that Г is nilpotent iff either of these two central series is finite and the class n of Г is the number of nontrivial subgroups in either central series.

Keywords

Riemann Surface Fundamental Group Fundamental Domain Discrete Subgroup Kleinian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Boston 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, DavisDavisU.S.A.

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