Hyperbolic Manifolds and Discrete Groups pp 333-350 | Cite as

# Brooks’ Theorem and Circle Packings

## Abstract

Which geometrically finite subgroups *G* of Isom(H^{3}) are contained in the lattices Г in Isom(H^{3})? It is clear that there are only countably many lattices Г and there is a continuum of geometrically finite subgroups *G* ⊂ Isom(H^{3}). In this chapter, we present a theorem of R. Brooks that asserts that in some sense geometrically finite subgroups *G* ⊂ Isom(H^{3}), which could be extended to lattices, form a dense countable set. Below is an outline of the proof. Suppose that *P* is a finite collection of closed round disks in the 2-sphere *S* ^{2}2 such that the interiors of distinct disks are disjoint; however, the boundary circles of these disks could be tangent. Such a collection of circles is a partial packing of *S* ^{2}. A partial *packing* is a packing if each complementary component T to the union of disks in *P* is an *ideal triangle*, i.e., the boundary of *T* consists of three circular arcs. Given a partial packing V, one may ask if it is possible to extend *P* to a packing *Q* by adding more round disks to the complementary components of ∪_{DϵP } *D*. One can easily add more disks to *P* and get a partial packing *P*, where each complementary component is an ideal quadrilateral. However, in general, we may be unable to extend *P*’ further, to a finite packing. We could keep adding more and more disks to each complementary quadrilateral

## Keywords

Fundamental Domain Kleinian Group Circle Packing Continuous Fraction Expansion Finite Subgroup## Preview

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