Bicuspid F ‐structures and Hecke groups

Viewing the Teichmüller disk determined by a meromorphic quadratic differential q ∈ T *(( p , n )), n (3 p −3 + n )>0, as a family of flat cone metrics on a fixed punctured surface, the level sets of the function ‘isotopy class of Delaunay partition’ tessellate the Poincaré disk by geodesic polygons of finite area and, possibly, one or more ideal vertices (cusps). At least one tile has a cusp if and only if, the differential admits a purely periodic (Strebel) direction. If two differentials are related by a chain of coverings, each unbranched away from the zero sets, they determine the same tessellation. Every open tile has exactly two cusps if and only if: (a) for some n >4, every tile is a Poincaré triangle with angles 0,0 and 2π/ n and (b) after perhaps passing to a ℤ 2 ‐extension, one flat surface in the Teichmüller disk covers the flat surface obtained from gluing opposite parallel edges of a regular n ‐gon P n , n even, or the union P n ∪ P n * of a regular n ‐gon with its reflection in an edge, n odd. The covering in (b) is unbranched away from the vertex class(es).