New Cone Metrics on the Sphere

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We give an explicit construction of lattices in P U (1, 2). A family of these lattices was originally constructed by Livn´e [15]. Parker [19] constructed these lattices of Livn´e as the modular group of certain Euclidean cone metrics on the sphere. In this work we give a construction of these lattices which includes that of Parker’s as the modular group of certain Euclidean cone metrics on the sphere. Our cone metrics on the sphere had five cone points with cone angles (π − θ + 2φ, π + θ, π + θ, π + θ, 2π − 2θ − 2φ) Where θ > 0, φ > 0 and θ + φ < π. These corresponds to a group of five tuples lattices generated by Thurston [27] in his paper Shapes of Polyhedra and Triangulations of the Sphere . Hence our choice of θ and φ in order to obtain discreteness are as follows: θ 2π/3 2π/3 2π/3 2π/4 2π/4 (2π/5) 2π/5 2π/6 φ π/4 π/5 π/6 π/3 π/4 (2π/5) π/3 π/3 Certain automorphisms which we considered on our cone metrics yielded unitary matrices R1, R2 and I1. Using these matrices, we obtained our fundamental polyhe- dron D by constructing our vertices, edges and faces to define the polyhedron. Our vertices were obtained by the degeneration of certain cone metrics. The polyhedron D is contained in bisectors whose intersection give us the edges of the polyheron. The faces are also contained in the bisectors. Then finally we proved using Poincar´e’s polyhedron theorem that the group Γ generated by the side pairings of D is a dis- crete subgroup of P U (1, 2) with fundamental domain D and presentation: J 3 = Rp = Rp = (P −1J )k = I , \ 1 2 Γ = J, P, R1, R2 : R2 = P R1P −1 = J R1J −1, P = R1R2
A Thesis submitted to the Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi in partial fulfillment of the requirements for the degree of Doctor of Philosophy, June-2011