Hyperbolic Manifolds and Discrete Groups pp 57-118 | Cite as

# Kleinian Groups

Chapter

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## Abstract

Our discussion of nilpotent groups follows [War76]. If
Where each Г
The smallest number

*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$$ \cdots \subset \Gamma _3 \subset \Gamma _2 \subset \Gamma _1 = \Gamma ,$$

_{ 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$$1 = \Gamma _{n + 1} \subset \cdots \subset \Gamma _3 \subset \Gamma _2 \subset \Gamma _1 = \Gamma .$$

*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 }:=Z_{i}(Г) are defined as follows. Z_{ n }(Г) is the center of Г. Inductively, Z_{i–1}(Г) :={x ϵ Г : xZ_{i}(Г)) ϵ center (Г/Z_{i}(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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