Geometry & Topology

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

Francesco Costantino and Bruno Martelli

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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.

Article information

Geom. Topol., Volume 18, Number 3 (2014), 1485-1538.

Received: 13 June 2013
Accepted: 15 January 2014
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57R56: Topological quantum field theories
Secondary: 57M27: Invariants of knots and 3-manifolds 22D10: Unitary representations of locally compact groups

quantum invariants mapping class groups representations in Hilbert space


Costantino, Francesco; Martelli, Bruno. An analytic family of representations for the mapping class group of punctured surfaces. Geom. Topol. 18 (2014), no. 3, 1485--1538. doi:10.2140/gt.2014.18.1485.

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