Geometry & Topology

Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

Francis Bonahon and Xiaobo Liu

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

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513799864

Digital Object Identifier
doi:10.2140/gt.2007.11.889

Mathematical Reviews number (MathSciNet)
MR2326938

Zentralblatt MATH identifier
1134.57008

Subjects
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]

Keywords
Quantum Teichmüller space surface diffeomorphisms

Citation

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. https://projecteuclid.org/euclid.gt/1513799864


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