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

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

2004-07-06

arxiv.org/abs/math/0407086v5

Mathematics

Geometric Topology

Revised introduction. To appear in Geometry & Topology

Scientific paper

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.

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