#### Volume 18, issue 3 (2014)

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

### Francesco Costantino and Bruno Martelli

Geometry & Topology 18 (2014) 1485–1538
##### 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 $|A|\le 1$ 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 $\left[-1,0\right]$ 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.

##### Keywords
quantum invariants, mapping class groups, representations in Hilbert space
##### Mathematical Subject Classification 2010
Primary: 57R56
Secondary: 57M27, 22D10