Brooks’ Theorem and Circle Packings

Abstract

Which geometrically finite subgroups G of Isom(H³) are contained in the lattices Г in Isom(H³)? It is clear that there are only countably many lattices Г and there is a continuum of geometrically finite subgroups G ⊂ Isom(H³). In this chapter, we present a theorem of R. Brooks that asserts that in some sense geometrically finite subgroups G ⊂ Isom(H³), 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 ²2 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