A Projective Representation of the Modular Group

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, $S{L}_2(\mathbb{Z})$. Also known as the modular group. We use this representation to associate a matrix to each element of $S{L}_2(\mathbb{Z})$; we then compute the trace and determinant of the associated matrix.

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