# Kleinian Groups

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
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