Universal Spin Teichmueller Theory, I. The action of P(SL(2,Z)) on Tess^+

TL;DR

This paper introduces the universal spin Teichmueller space and the spin mapping class group P(SL(2,Z)), demonstrating their properties and actions, extending classical Teichmueller theory to include spin structures.

Contribution

It defines the spin universal Teichmueller space Tess^+ and the group P(SL(2,Z)), showing their universality and providing explicit generators, extending the classical theory to spin structures.

Findings

P(SL(2,Z)) acts on Tess^+ universally for finite-type hyperbolic surfaces with spin.

Explicit generators of P(SL(2,Z)) are constructed and shown to generate the group.

A companion paper provides a finite presentation of P(SL(2,Z)).

Abstract

Earlier work took as universal mapping class group the collection PPSL(2,Z) of all piecewise PSL(2,Z) homeomorphisms of the unit circle S^1 with finitely many breakpoints among the rational points. The spin mapping class group P(SL(2,Z)) introduced here consists of all piecewise-constant maps from S^1 to SL(2,Z) which projectivize to an element of PPSL(2,Z). We also introduce a spin universal Teichmueller space Tess^+ covering the earlier universal Teichmueller space Tess of tesselations of the Poincare disk D with fiber the space of Z/2 connections on the graphs dual to the tesselations in D. There is a natural action of P(SL(2,Z)) on Tess^+ which is universal for finite-type hyperbolic surfaces with spin structure in the same sense that the action of PPSL(2,Z) on Tess is universal for finite-type hyperbolic surfaces. Three explicit elements of P(SL(2,Z)) are defined combinatorially via their actions on Tess^+, and the main new result here is that they generate P(SL(2,Z)). Background, including material on hyperbolic and spin structures on finite-type surfaces, is sketched down to first principles in order to motivate the new constructions and to provide an overall survey. A companion paper to this one gives a finite presentationof the universal spin mapping class group P(SL(2,Z)) introduced here.