A deformation of Penner's simplicial coordinate

Abstract

We find a one-parameter family of coordinates $\{\Psi_h\}_{h\in\mathbb{R}}$ 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 $h \ge 0$, the decorated Teichmüller space in the $\Psi_h$ 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 $P_h(T)$ so that $P_h(T)\subset P_{h'} (T)$ if $h < h'$. As a consequence, Bowditch-Epstein and Penner's cell decomposition of the decorated Teichmüller space is reproduced.