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An analytic family of representations for the mapping class group of punctured surfaces

Francesco Costantino and Bruno Martelli

Geometry & Topology 18 (2014) 1485–1538

We use quantum invariants to define an analytic family of representations for the mapping class group Mod(Σ) of a punctured surface Σ. The representations depend on a complex number A with |A| 1 and act on an infinite-dimensional Hilbert space. They are unitary when A is real or imaginary, bounded when |A| < 1, and only densely defined when |A| = 1 and A is not a root of unity. When A is a root of unity distinct from ± 1 and ± i the representations are finite-dimensional and isomorphic to the “Hom” version of the well-known TQFT quantum representations.

The unitary representations in the interval [1,0] interpolate analytically between two natural geometric unitary representations, the SU(2)–character variety representation studied by Goldman and the multicurve representation induced by the action of Mod(Σ) on multicurves.

The finite-dimensional representations converge analytically to the infinite-dimensional ones. We recover Marché and Narimannejad’s convergence theorem, and Andersen, Freedman, Walker and Wang’s asymptotic faithfulness, that states that the image of a noncentral mapping class is always nontrivial after some level r0. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r0 in terms of its dilatation.

quantum invariants, mapping class groups, representations in Hilbert space
Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 57M27, 22D10
Received: 13 June 2013
Accepted: 15 January 2014
Published: 7 July 2014
Proposed: Walter Neumann
Seconded: Danny Calegari, Jean-Pierre Otal
Francesco Costantino
Institut de Recherche Mathématique Avancée
Rue René Descartes 7
67087 Strasbourg
Bruno Martelli
Dipartimento di Matematica
Università di Pisa
Largo Pontecorvo 5
I-56127 Pisa