Volume 11, issue 2 (2007)

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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
 arXiv: math.GT/0407086
Abstract

We investigate the representation theory of the polynomial core ${\mathsc{T}}_{S}^{q}$ 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 ${\mathsc{T}}_{S}^{q}$ are classified, up to finitely many choices, by group homomorphisms from the fundamental group ${\pi }_{1}\left(S\right)$ 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