Brooks’ Theorem and Circle Packings
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
KeywordsFundamental Domain Kleinian Group Circle Packing Continuous Fraction Expansion Finite Subgroup
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