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.
An analytic family of representations for the mapping class group of punctured surfaces
MARTELLI, BRUNO
2014-01-01
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.File | Dimensione | Formato | |
---|---|---|---|
1210.2666.pdf
accesso aperto
Descrizione: Articolo in forma postprint
Tipologia:
Documento in Post-print
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
793.01 kB
Formato
Adobe PDF
|
793.01 kB | Adobe PDF | Visualizza/Apri |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.