## Geometry & Topology

### 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 $Mod(Σ)$ 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.

#### Article information

Source
Geom. Topol., Volume 18, Number 3 (2014), 1485-1538.

Dates
Accepted: 15 January 2014
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513732800

Digital Object Identifier
doi:10.2140/gt.2014.18.1485

Mathematical Reviews number (MathSciNet)
MR3228457

Zentralblatt MATH identifier
1311.57041

#### Citation

Costantino, Francesco; Martelli, Bruno. An analytic family of representations for the mapping class group of punctured surfaces. Geom. Topol. 18 (2014), no. 3, 1485--1538. doi:10.2140/gt.2014.18.1485. https://projecteuclid.org/euclid.gt/1513732800

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