## Geometry & Topology

### Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms

#### 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 $S$.

#### Article information

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

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

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

#### 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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