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Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Francis Bonahon and Xiaobo Liu

Geometry & Topology 11 (2007) 889–937

DOI: 10.2140/gt.2007.11.889

arXiv: math.GT/0407086
Abstract

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.

Keywords

Quantum Teichmüller space, surface diffeomorphisms
Mathematical Subject Classification

Primary: 57R56

Secondary: 20G42, 57M50
References
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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
Authors
Francis Bonahon
Department of Mathematics
University of Southern California
Los Angeles
CA 90089-2532
USA
Xiaobo Liu
Department of Mathematics
Columbia University
2990 Broadway
New York, NY 10027
USA