Geometric Galois Actions

Introduction

A new model of a universal Teichmüller space was introduced in [P2], and a universal analogue of the mapping class groups, called the “universal Ptolemy group”, was denned and studied. In our concentration on the geometry in [P2], we perhaps obfuscated the essentially easy algebraic and combinatorial arguments underlying this definition, and one goal here is to give a gentle survey of these aspects and a complete definition of the universal Ptolemy group. To be sure, the geometric side of the story provides the main calculational tools of the theory (and explains the terminology “Ptolemy group”), but we do not discuss these aspects here.

To hopefully gain some perspective on the results in this paper, we next review the “universal (decorated) Teichmüller theory” from [P2]. In this context, the appellate “universal” means that we seek certain infinite dimensional spaces (actually, we shall find infinite–dimensional symplectic Fréchet manifolds supporting a group action) together with maps of each of the “classical (decorated) Teichmüller spaces” into our universal object in such a way that classical combinatorial, topological, and geometric structures arise as the pull–back or restriction of corresponding (group invariant) structures on the universal objects.

The classical “(decorated) Teichmüller theory” may be described as follows.