CROSSED HOMOMORPHISMS AND LOW DIMENSIONAL REPRESENTATIONS OF MAPPING CLASS GROUPS OF SURFACES

We continue the study of low dimensional linear representations of mapping class groups of surfaces initiated by Franks-Handel [Proc. Amer. Math. So. 141 (2013), pp. 2951-2962] and Korkmaz [Low-dimensional linear representations of mapping class groups, preprint, arXiv:1104.4816v2 (2011)]. We consider (2g + 1)-dimensional complex linear representations of the pure mapping class groups of compact orientable surfaces of genus g. We give a complete classification of such representations for g > 7 up to conjugation, in terms of certain twisted 1-cohomology groups of the mapping class groups. A new ingredient is to use the computation of a related twisted 1-cohomology group by Morita [Ann. Inst. Fourier (Grenoble) 39 (1989), pp. 777-810]. The classification result implies in particular that there are no irreducible linear representations of dimension 2g + 1 for g >= 7, which marks a contrast with the case g = 2.