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

Costantino, Francesco; Martelli, Bruno

doi:10.2140/gt.2014.18.1485

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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

DOI: 10.2140/gt.2014.18.1485

Abstract

We use quantum invariants to define an analytic family of representations for the mapping class group $Mod (Σ)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Mod</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>Σ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ of a punctured surface $Σ$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>Σ</mi></math>$ . The representations depend on a complex number $A$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ with $| A | \leq 1$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-rel">|</mo><mi>A</mi><mo class="MathClass-rel">|</mo><mo class="MathClass-rel">≤</mo> <mn>1</mn></math>$ and act on an infinite-dimensional Hilbert space. They are unitary when $A$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ is real or imaginary, bounded when $| A | < 1$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-rel">|</mo><mi>A</mi><mo class="MathClass-rel">|</mo> <mo class="MathClass-rel">&lt;</mo> <mn>1</mn></math>$ , and only densely defined when $| A | = 1$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="MathClass-rel">|</mo><mi>A</mi><mo class="MathClass-rel">|</mo> <mo class="MathClass-rel">=</mo> <mn>1</mn></math>$ and $A$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ is not a root of unity. When $A$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>A</mi></math>$ is a root of unity distinct from $\pm 1$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mo class="MathClass-bin">±</mo> <mn>1</mn></math>$ and $\pm i$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <mo class="MathClass-bin">±</mo> <mi>i</mi></math>$ the representations are finite-dimensional and isomorphic to the “Hom” version of the well-known TQFT quantum representations.

The unitary representations in the interval $[- 1, 0]$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mo class="MathClass-open">[</mo><mrow><mo class="MathClass-bin">−</mo><mn>1</mn><mo class="MathClass-punc">,</mo><mn>0</mn></mrow><mo class="MathClass-close">]</mo></mrow></math>$ interpolate analytically between two natural geometric unitary representations, the $SU (2)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> SU</mo><mrow><mo class="MathClass-open">(</mo><mrow><mn>2</mn></mrow><mo class="MathClass-close">)</mo></mrow></math>$ –character variety representation studied by Goldman and the multicurve representation induced by the action of $Mod (Σ)$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mo class="qopname"> Mod</mo><mrow><mo class="MathClass-open">(</mo><mrow><mi>Σ</mi></mrow><mo class="MathClass-close">)</mo></mrow></math>$ 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 $r_{0}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math>$ . When the mapping class is pseudo-Anosov we give a simple polynomial estimate of the level $r_{0}$ $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><msub><mrow><mi>r</mi></mrow><mrow><mn>0</mn></mrow></msub></math>$ in terms of its dilatation.

Keywords

quantum invariants, mapping class groups, representations in Hilbert space

Mathematical Subject Classification 2010

Primary: 57R56

Secondary: 57M27, 22D10

References

Publication

Received: 13 June 2013

Accepted: 15 January 2014

Published: 7 July 2014

Proposed: Walter Neumann

Seconded: Danny Calegari, Jean-Pierre Otal

Authors

Francesco Costantino
Institut de Recherche Mathématique Avancée Rue René Descartes 7 67087 Strasbourg France

Bruno Martelli
Dipartimento di Matematica Università di Pisa Largo Pontecorvo 5 I-56127 Pisa Italy

Volume 18, issue 3 (2014)