On the number of optimal surfaces

Let X be a closed oriented Riemann surface of genus > 1 of constant negative curvature -1. A surface containing a disk of maximal radius is an optimal surface. This paper gives exact formulae for the number of optimal surfaces of genus > 3 up to orientation-preserving isometry. We show that the automorphism group of such a surface is always cyclic of order 1,2,3 or 6. We also describe a combinatorial structure of nonorientable hyperbolic optimal surfaces.Comment: This is the version published by Geometry & Topology Monographs on 29 April 200

Polyhedra with specified links

We construct compact polyhedra with $m$-gonal faces whose links are generalized 3-gons. It gives examples of cocompact hyperbolic bildings of type $P(m,3)$. For $m=3$ we get compact spaces covered by Euclidean buildings of type $A_2$

Triangular hyperbolic buildings

We construct triangular hyperbolic polyhedra whose links are generalized 4-gons. The universal cover of those polyhedra are hyperbolic buildings, which appartments are hyperbolic planes tesselated by regular triangles with angles $\pi/4$. Moreover, the fundamental groups of the polyhedra acts simply transitively on vertices of the buildings