Paper detail

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 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.