A deformation of Penner's simplicial coordinate

TL;DR

This paper introduces a one-parameter family of coordinates deforming Penner's simplicial coordinate on decorated Teichmüller space, revealing convex polytopes and reproducing known cell decompositions.

Contribution

It constructs a new deformation of Penner's coordinate system, providing explicit convex polytopes for different parameter values and connecting to existing cell decompositions.

Findings

For h ≥ 0, the coordinates form a convex polytope P(T).

For h < 0, the coordinates form a bounded convex polytope P_h(T).

The deformation reproduces Bowditch-Epstein and Penner's cell decompositions.

Abstract

We produce a one-parameter family of coordinates $\{\Psi_h\}_{h\in\mathbb{R}}$ of the decorated Teichm\"{u}ller space of an ideally triangulated punctured surface $(S,T)$ with negative Euler characteristic, which is a deformation of Penner's simplicial coordinate \cite{P1}. If $h\geqslant0$, the decorated Teichm\"{u}ller space in the $\Psi_h$ coordinate becomes an explicit convex polytope $P(T)$ independent of $h$, and if $h<0$, the decorated Teichm\"{u}ller 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 Teichm\"{u}ller space is reproduced.