Geometry & Topology

Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Francis Bonahon and Xiaobo Liu

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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 3. We exploit this connection between algebra and hyperbolic geometry to exhibit invariants of diffeomorphisms of S.

Article information

Geom. Topol., Volume 11, Number 2 (2007), 889-937.

Received: 16 December 2005
Accepted: 13 December 2006
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R56: Topological quantum field theories
Secondary: 57M50: Geometric structures on low-dimensional manifolds 20G42: Quantum groups (quantized function algebras) and their representations [See also 16T20, 17B37, 81R50]

Quantum Teichmüller space surface diffeomorphisms


Bonahon, Francis; Liu, Xiaobo. Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol. 11 (2007), no. 2, 889--937. doi:10.2140/gt.2007.11.889.

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