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IMAGES OF QUANTUM REPRESENTATIONS OF MAPPING CLASS GROUPS AND DUPONT–GUICHARDET–WIGNER QUASI-HOMOMORPHISMS

Part of: Low-dimensional topology Special aspects of infinite or finite groups Steinberg groups and $K_2$ Homology and cohomology theories

Published online by Cambridge University Press:  27 January 2016

Louis Funar  and
Wolfgang Pitsch
Louis Funar
Affiliation:
Institut Fourier, UMR 5582, Mathématiques, University Grenoble Alpes, CS 40700, 38058 Grenoble cedex 9, France (louis.funar@univ-grenoble-alpes.fr)
Wolfgang Pitsch
Affiliation:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Cerdanyola del Vallès), Espana (pitsch@mat.uab.es)
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Abstract

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further, we show that the images of the mapping class groups have non-trivial 2-cohomology, at least for small levels. For this purpose, we considered a series of quasi-homomorphisms on mapping class groups extending the previous work of Barge and Ghys (Math. Ann. 294 (1992), 235–265) and of Gambaudo and Ghys (Bull. Soc. Math. France 133(4) (2005), 541–579). These quasi-homomorphisms are pull-backs of the Dupont–Guichardet–Wigner quasi-homomorphisms on pseudo-unitary groups along quantum representations.

Keywords

symplectic grouppseudo-unitary groupDupont–Guichardet–Wigner cocyclequasi-homomorphismgroup homologymapping class groupcentral extensionquantum representation

MSC classification

Primary: 57M50: Geometric structures on low-dimensional manifolds 55N25: Homology with local coefficients, equivariant cohomology 19C09: Central extensions and Schur multipliers 20F38: Other groups related to topology or analysis
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 17 , Issue 2 , April 2018 , pp. 277 - 304
Copyright
© Cambridge University Press 2016 

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