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

Abstract

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 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, -1, i, -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 the mapping class group on multicurves. The finite-dimensional representations converge analytically to the infinite-dimensional ones. We recover Marche and Narimannejad's convergence theorem, and Andersen, Freedman, Walker and Wang's asymptotic faithfulness, that states that the image of a non-central mapping class is always non-trivial after some level r. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r in term of its dilatation.