Cone metrics on the sphere and Livné's lattices (Q2500465)

In his dissertation, R. Livné constructed a family of lattices in \(\text{PU}(1,2)\), the isometries of complex \(2\)-dimensional hyperbolic space. An alternative construction was given by \textit{W. Thurston} [Shapes of polyhedra and triangulations of the sphere, The Epstein Birthday Schrift dedicated to David Epstein on the occasion of his 60th birthday. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 1, 511--549 (1998; Zbl 0931.57010)]. His method was to consider Euclidean cone metrics of the \(2\)-sphere with five cone points and certain prescribed cone angles. An automorphism of the cone metric gives rise to a unitary matrix in \(\text{U}(1,2)\). Each similarity class of cone metrics corresponds to a positive point in \(\text{\textbf{P}}(\text{\textbf{C}}^{1,2})\), that is, a point in complex hyperbolic space, and the automorphisms of similarity classes correspond to complex hyperbolic isometries. Under certain conditions, the groups \(\Gamma\) generated by these automorphisms turn out to be Livné lattices. In this paper, the author reworks Thurston's approach in considerable depth. Explicit fundamental domains for the actions on complex hyperbolic space are constructed and completely understood. Discreteness of the groups is proven using the Poincaré polyhedron theorem, yielding also simple finite presentations for the groups. Additional presentations are obtained that reveal other structure of the groups; in particular, that they contain triangle groups as normal subgroups. The paper is carefully written, and noteworthy for its use of explicit computation while keeping the underlying geometry in view.