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G&T Monographs |
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Representations of the quantum Teichmüller space and
invariants of surface diffeomorphisms
Francis Bonahon and Xiaobo Liu
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Geometry & Topology 11 (2007)
889–937
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Abstract
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We investigate
the representation theory of the polynomial core TSq 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
TSq are classified, up to finitely many choices, by group
homomorphisms from the fundamental group π1(S) to the isometry
group of the hyperbolic 3–space H3. We exploit this
connection between algebra and hyperbolic geometry to exhibit
invariants of diffeomorphisms of S.
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Keywords
Quantum Teichmüller space, surface
diffeomorphisms
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Mathematical Subject Classification
Primary: 57R56
Secondary: 20G42, 57M50
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Publication
Received: 16 December 2005
Accepted: 13 December 2006
Published: 27 May 2007
Proposed: Jean-Pierre Otal
Seconded: Walter Neumann, Joan Birman
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