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

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.