Universal Spin Teichmüller Theory, I: The Action of P(SL(2, ℤ $$\mathbb {Z}$$ )) on T ess + $$\mathcal {T}\mathrm {ess}^+$$

Earlier work took as universal mapping class group the collection PPSL ( 2 , ℤ ) $$\mathrm {PPSL}(2,{\mathbb Z})$$ of all piecewise PSL ( 2 , ℤ ) $$\mathrm {PSL}(2,{\mathbb Z})$$ homeomorphisms of the unit circle S 1 = ∂ 𝔻 $$S^1=\partial {\mathbb D}$$ with finitely many breakpoints among the rational points in S 1 $$S^1$$ . The spin mapping class group P(SL(2, ℤ $${\mathbb Z}$$ )) introduced here consists of all piecewise-constant maps S 1 → SL ( 2 , ℤ ) $$S^1\to \mathrm {SL}(2,{\mathbb Z})$$ which projectivize to an element of PPSL ( 2 , ℤ ) $$\mathrm {PPSL}(2,{\mathbb Z})$$ . We also introduce a spin universal Teichmüller space T es s + $$\mathcal {T}ess^+$$ covering the earlier universal Teichmüller space T ess $$\mathcal {T}ess$$ of tesselations of 𝔻 $${\mathbb D}$$ with fiber the space of ℤ $${\mathbb Z}$$ /2 connections on the graph dual to the tesselation in 𝔻 $${\mathbb D}$$ . There is a natural action The image contains a mathematical formula: \( \mathbb{P}(\text{SL}(2, \mathbb{Z})) \hookrightarrow \mathcal{T}_{\text{ess}}^+ \). The formula includes symbols such as the set of integers \(\mathbb{Z}\), the projective special linear group \(\text{SL}(2, \mathbb{Z})\), and the essential tessellation space \(\mathcal{T}_{\text{ess}}^+\). The arrow \(\hookrightarrow\) indicates an embedding or inclusion. which is universal for finite-type hyperbolic surfaces with spin structure in the same sense that The image contains a mathematical notation: "PPSL(2, \mathbb{Z}) \hookrightarrow \mathcal{T}_{ess}". It includes symbols such as the set of integers \(\mathbb{Z}\), a hook arrow indicating an embedding or injection, and script-style letters for "Tess". is universal for finite-type hyperbolic surfaces. Three explicit elements of P(SL(2, ℤ $${\mathbb Z}$$ )) are defined combinatorially via their actions on T es s + $$\mathcal {T}ess^+$$ , and the main new result here is that they generate P(SL(2, ℤ $${\mathbb Z}$$ )). 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 chapter to this one gives a finite presentation of the universal spin mapping class group P ( SL ( 2 , ℤ ) ) $$\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))$$ introduced here.