Geometry & Topology
- Geom. Topol.
- Volume 18, Number 3 (2014), 1485-1538.
An analytic family of representations for the mapping class group of punctured surfaces
We use quantum invariants to define an analytic family of representations for the mapping class group of a punctured surface . The representations depend on a complex number with and act on an infinite-dimensional Hilbert space. They are unitary when is real or imaginary, bounded when , and only densely defined when and is not a root of unity. When is a root of unity distinct from and the representations are finite-dimensional and isomorphic to the “Hom” version of the well-known TQFT quantum representations.
The unitary representations in the interval interpolate analytically between two natural geometric unitary representations, the –character variety representation studied by Goldman and the multicurve representation induced by the action of 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 . When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level in terms of its dilatation.
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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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. https://projecteuclid.org/euclid.gt/1513732800