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 $Mod\left(\Sigma \right)$ of a punctured surface $\Sigma$. The representations depend on a complex number $A$ with $\left|A\right|\le 1$ and act on an infinite-dimensional Hilbert space. They are unitary when $A$ is real or imaginary, bounded when $\left|A\right|<1$, and only densely defined when $\left|A\right|=1$ and $A$ is not a root of unity. When $A$ is a root of unity distinct from $\pm 1$ and $\pm i$ the representations are finite-dimensional and isomorphic to the “Hom” version of the well-known TQFT quantum representations.

The unitary representations in the interval $\left[-1,0\right]$ interpolate analytically between two natural geometric unitary representations, the $SU\left(2\right)$–character variety representation studied by Goldman and the multicurve representation induced by the action of $Mod\left(\Sigma \right)$ 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 $r_0$. When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level $r_0$ in terms of its dilatation.