Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Abstract

We investigate the representation theory of the polynomial core ${\mathcal{T}}_S^q$ of the quantum Teichmüller space of a punctured surface $S$. This is a purely algebraic object, closely related to the combinatorics of the simplicial complex of ideal cell decompositions of $S$. Our main result is that irreducible finite-dimensional representations of ${\mathcal{T}}_S^q$ are classified, up to finitely many choices, by group homomorphisms from the fundamental group ${\pi }_1\left(S\right)$ to the isometry group of the hyperbolic 3–space ${ℍ}^3$. We exploit this connection between algebra and hyperbolic geometry to exhibit invariants of diffeomorphisms of $S$.