A Projective Representation of the Modular Group

TL;DR

This paper explores a projective representation of the modular group derived from quantum Teichmuller theory, analyzing associated matrices' traces and determinants to understand invariants of three-manifolds.

Contribution

It introduces a fixed-point representation of the modular group from quantum Teichmuller theory and computes associated matrix invariants.

Findings

Computed traces and determinants of matrices associated with the modular group.

Established a fixed-point representation invariant under all modular group elements.

Linked the representation to quantum invariants of three-manifolds.

Abstract

Quantum Teichmuller theory assigns invariants to three-manifolds via projective representations of mapping class groups derived from the representation of a noncommutative torus. Here, we focus on a representation of the simplest non-commutative torus which remains fixed by all elements of the mapping class group of the torus, $SL_2(\mathbb{Z})$. Also known as the modular group. We use this representation to associate a matrix to each element of $SL_2(\mathbb{Z})$; we then compute the trace and determinant of the associated matrix.