Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Bonahon, Francis; Liu, Xiaobo

doi:10.2140/gt.2007.11.889

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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 $T_{S}^{q}$ 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 $T_{S}^{q}$ are classified, up to finitely many choices, by group homomorphisms from the fundamental group $π_{1} (S)$ 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 $S$ .

Keywords

Quantum Teichmüller space, surface diffeomorphisms

Mathematical Subject Classification 2000

Primary: 57R56

Secondary: 57M50, 20G42

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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

Volume 11, issue 2 (2007)