Universal Spin Teichmüller Theory, II: Finite Presentation of P(SL(2, ℤ $${\mathbb Z}$$ ))

In previous works, the universal mapping class group was taken to be the group 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 $$S^1$$ with finitely many breakpoints among the rational points in S 1 $$S^1$$ , and in fact, the Thompson group T ≈ PPSL ( 2 , ℤ ) $$T\approx {\mathrm {PPSL}}(2,{\mathbb Z})$$ . The new spin mapping class group P(SL(2, ℤ $${\mathbb Z}$$ )) is given by 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 compute a finite presentation of PPSL ( 2 , ℤ ) $${\mathrm {PPSL}}(2,{\mathbb Z})$$ from basic principles of general position as an orbifold fundamental group. The orbifold deck group of the spin cover is explicitly computed here, from which follows also a finite presentation of P ( SL ( 2 , ℤ ) ) $$\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))$$ . This is our main new achievement. Certain commutator relations in P ( SL ( 2 , ℤ ) ) $$\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))$$ seem to organize according to root lattices, which would be a novel development. We naturally wonder what is the automorphism group of P ( SL ( 2 , ℤ ) ) $$\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))$$ and speculate that it is a large sporadic group. There is a companion chapter to this one which explains the topological background from first principles and proves that the group studied here using combinatorial group theory is indeed P ( SL ( 2 , ℤ ) ) $$\mathrm {P}(\mathrm {SL}(2,{\mathbb Z}))$$ .