By Mumford D Tata

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**Extra resources for Lectures on Theta II Birkhaeuser**

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1. Let Γ be an integral Kleinian group of isometries of Hn of ﬁnite covolume, for example, the orthogonal group of some integral quadratic lattice L. It is obviously geometrically ﬁnite, however, for n > 3 it may contain ﬁnitely generated subgroups which are not geometrically ﬁnite. 9 A sphere packing P = (Si )i∈I is called integral if the following conditions are satisﬁed. (i) The corresponding norm one vectors vi in Rn+1,1 span an integral lattice L; (ii) There exists a positive integer λ such that λ(vi , vj ) ∈ Z for all i, j ∈ I.

8From a “poem proof” of the theorem in the case n = 2 in “Kiss Precise” by Frederick Soddy published in Nature, 1930: Four circles to the kissing come. / The smaller are the benter. / The bend is just the inverse of / The distance from the center. / Though their intrigue left Euclid dumb / There’s now no need for rule of thumb. / Since zero bend’s a dead straight line / And concave bends have minus sign, / The sum of the squares of all four bends / Is half the square of their sum. 36 IGOR DOLGACHEV Figure 2.

9 A sphere packing P = (Si )i∈I is called integral if the following conditions are satisﬁed. (i) The corresponding norm one vectors vi in Rn+1,1 span an integral lattice L; (ii) There exists a positive integer λ such that λ(vi , vj ) ∈ Z for all i, j ∈ I. The smallest such c is called the exponent. (iii) The curvatures of spheres Si are integers. After multiplying the quadratic form of L, the restriction of the fundamental quadratic form to L, by the exponent we obtain an integral quadratic lattice int(L).