#### Volume 13, issue 6 (2013)

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Factorization rules in quantum Teichmüller theory

### Julien Roger

Algebraic & Geometric Topology 13 (2013) 3411–3446
##### Abstract

For a punctured surface $S$, a point of its Teichmüller space $\mathsc{T}\left(S\right)$ determines an irreducible representation of its quantization ${\mathsc{T}}^{q}\left(S\right)$. We analyze the behavior of these representations as one goes to infinity in $\mathsc{T}\left(S\right)$, or in the moduli space $\mathsc{ℳ}\left(S\right)$ of the surface. The main result of this paper states that an irreducible representation of ${\mathsc{T}}^{q}\left(S\right)$ limits to a direct sum of representations of ${\mathsc{T}}^{q}\left({S}_{\gamma }\right)$, where ${S}_{\gamma }$ is obtained from $S$ by pinching a multicurve $\gamma$ to a set of nodes. The result is analogous to the factorization rule found in conformal field theory.

##### Keywords
quantum Teichmüller space, Weil–Petersson geometry, ideal triangulations, shear coordinates
##### Mathematical Subject Classification 2010
Primary: 57M50
Secondary: 32G15, 20G42
##### Publication
Received: 8 February 2013
Accepted: 19 April 2013
Published: 10 October 2013
##### Authors
 Julien Roger Department of Mathematics Rutgers University 110 Frelinghuysen Rd Piscataway, NJ 08854-8019 USA http://math.rutgers.edu/~juroger