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 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.