Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Bonahon, Francis¹ ; Liu, Xiaobo²

¹ Department of Mathematics, University of Southern California, Los Angeles, CA 90089-2532, USA
² Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA

Geometry & topology, Tome 11 (2007) no. 2, pp. 889-937.

Résumé

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 $S$.

Détail

DOI : 10.2140/gt.2007.11.889

Keywords: Quantum Teichmüller space, surface diffeomorphisms

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