A deformation of penner’s simplicial coordinate

Abstract

We find a one-parameter family of coordinates (Ψh)h∈R which is a deformation of Penner’s simplicial coordinate of the decorated Teichmüller space of an ideally triangulated punctured surface (S; T) of negative Euler characteristic. If h ≥ 0, the decorated Teichmüller space in the Ψh coordinate becomes an explicit convex polytope P(T) independent of h, and if h < 0, the decorated Teichmüller space becomes an explicit bounded convex polytope Ph(T) so that Ph(T) ⊂ Ph’(T) if h < hh’. As a consequence, Bowditch-Epstein and Penner’s cell decomposition of the decorated Teichmüller space is reproduced. © 2011 J. Differential Geometry.