Universal Spin Teichmueller Theory, II. Finite Presentation of P(SL(2,Z))

TL;DR

This paper provides a finite presentation of the spin mapping class group P(SL(2,Z)), extending previous work on PPSL(2,Z), and explores its algebraic structure and potential automorphisms.

Contribution

It computes a finite presentation of P(SL(2,Z)) from fundamental principles, including the orbifold deck group, and investigates its algebraic organization.

Findings

Finite presentation of PPSL(2,Z) derived from basic principles.

Explicit finite presentation of P(SL(2,Z)) obtained.

Potential automorphism group of P(SL(2,Z)) may be a large sporadic group.

Abstract

In previous works, the universal mapping class group was taken to be the group PPSL(2,Z) of all piecewise PSL(2,Z) homeomorphisms of the unit circle S^1 with finitely many breakpoints among the rational points, and in fact, the Thompson group T is isomorphic to PPSL(2,Z). The new spin mapping class group P(SL(2,Z)) is defined to be all piecewise-constant maps from S^1 to SL(2,Z) which projectivize to an element of PPSL(2,Z). We compute a finite presentation of PPSL(2,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)). This is our main new achievement. Certain commutator relations in P(SL(2,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)) and speculate that it is a large sporadic group. There is a companion paper to this one which explains the topological background from first principles, proves that the group studied here using combinatorial group theory is indeed P(SL(2,Z)).