Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces

Kasahara, Yasushi

doi:10.1090/tran/9037

Crossed homomorphisms and low dimensional representations of mapping class groups of surfaces
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by Yasushi Kasahara;

Trans. Amer. Math. Soc. 377 (2024), 1183-1218

DOI: https://doi.org/10.1090/tran/9037

Published electronically: October 11, 2023

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Abstract:

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 $g \geq 7$ 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 $g \geq 7$, which marks a contrast with the case $g=2$.

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