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

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.