Representations of the quantum Teichmuller space, and invariants of surface diffeomorphisms

Abstract

We investigate the representation theory of the polynomial core of the quantum Teichmuller 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 this polynomial core are classified, up to finitely many choices, by group homomorphisms from the fundamental group of the surface to the isometry group of the hyperbolic 3--space. We exploit this connection between algebra and hyperbolic geometry to exhibit new invariants of diffeomorphisms of S.

Revised introduction. To appear in Geometry & Topology

References in corpus (1)

• Quantum hyperbolic invariants for diffeomorphisms of small surfaces

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