The mapping class group acts reducibly on 𝑆𝑈(𝑛)-character varieties

Goldman, William M.

doi:10.1090/conm/416/07889

Primes and Knots

2006

DOI: 10.1090/conm/416/07889

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The mapping class group acts reducibly on 𝑆𝑈(𝑛)-character varieties

William M. Goldman¹

Abstract: Let Σ be a compact orientable surface of genus g = 1 with n = 1 boundary component. The mapping class group Γ of Σ acts on the SU(3)-character variety of Σ. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

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2014

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

Costantino

Martelli

2014

Geom. Topol.

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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 infinitedimensional 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 non-central mapping class is always non-trivial after some level r 0 . When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level r 0 in term of its dilatation.

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

Costantino

Martelli

2014

Geom. Topol.

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The mapping class group acts reducibly on 𝑆𝑈(𝑛)-character varieties

Abstract: Let Σ be a compact orientable surface of genus g = 1 with n = 1 boundary component. The mapping class group Γ of Σ acts on the SU(3)-character variety of Σ. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

Cited by 1 publication

References 20 publications

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

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

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