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An analytic family of representations for the mapping class group of punctured surfaces. (English) Zbl 1311.57041
One application of quantum invariants is to construct interesting representations of mapping class groups. Such representations are the only rigourously constructed part of Witten’s gauge theoretical approach to Topological Quantum Field Theory (TQFT) in dimension 3. We are interested in their asymptotic properties.
In this breakthrough paper, the authors recover Marché and Narimannejad’s convergence theorem [J. Marché and M. Narimannejad, Duke Math. J. 141, No. 3, 573–587 (2008; Zbl 1139.57030)], see also [J. E. Andersen, Lett. Math. Phys. 91, No. 3, 205–214 (2010; Zbl 1195.57066)] and Andersen, Freedman, Walker, and Wang’s asymptotic faithfulness theorem [J. E. Andersen, Ann. Math. (2) 163, No. 1, 347–368 (2006; Zbl 1157.53049)], [M. H. Freedman et al., Geom. Topol. 6, 523–539 (2002; Zbl 1037.57024)] by revealing the relevant representations to be part of an analytic family of representations of mapping class groups of punctured surfaces. The above results then follow from examining families of representations which converge to the representation in question.
To be more specific, let $$\Sigma$$ denote a closed, oriented, connected surface with one or more punctures and with negative Euler characteristic if the punctures are removed. Let $$\mathbb{D}\subset \mathbb{C}$$ denote the unit disc, and let $$\bar{\mathbb{D}}$$ denote its closure. The authors construct a family $$\rho_A$$ of representations of the mapping class group $$\mathrm{Mod}(\Sigma)$$ on a subspace of the Hilbert space of multicurves for each $$A\in \bar{\mathbb{D}}$$. When $$A\in \mathbb{D}$$ the representations are bounded, when $$A$$ is real or imaginary the representations are unitary, and the representations are only densely defined when $$A$$ is on the unit circle but is not a root of unity. Important special cases are the $$\mathrm{SU}(2)$$ character variety representation for $$A=-1$$ and the multicurve representation induced by the action of $$\mathrm{Mod}(\Sigma)$$ on multicurves for $$A=0$$.
Asymptotic faithfulness states that the image of a non-central mapping class is always nontrivial after some level $$r_0$$. Theorem 1.7 gives a polynomial estimate of $$r_0$$ in terms of the mapping class’s dilation when the mapping class is pseudo-Anosov.
The main tools used are quantum 6j-symbols and A. Valette’s cocycle technique [C. R. Acad. Sci., Paris, Sér. I 310, No. 10, 703–708 (1990; Zbl 0828.22007)]. An important technical role is played by the combinatorics of the flip graph of triangulations of the surface with vertices at punctures.
Natural further questions include whether and how the results of the paper extend to surfaces without marked points, and how the representations split for various values of $$A$$.

MSC:
 57R56 Topological quantum field theories (aspects of differential topology) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 22D10 Unitary representations of locally compact groups
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