Low-dimensional linear representations of mapping class groups

Date

2023-09-01

Author

Korkmaz, Mustafa

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Let (Formula presented.) be a compact orientable surface of genus (Formula presented.) with marked points in the interior. Franks–Handel (Proc. Amer. Math. Soc. 141 (2013) 2951–2962) proved that if (Formula presented.) then the image of a homomorphism from the mapping class group (Formula presented.) of (Formula presented.) to (Formula presented.) is trivial if (Formula presented.) and is finite cyclic if (Formula presented.). The first result is our own proof of this fact. Our second main result shows that for (Formula presented.) up to conjugation there are only two homomorphisms from (Formula presented.) to (Formula presented.) : the trivial homomorphism and the standard symplectic representation. Our last main result shows that the mapping class group has no faithful linear representation in dimensions less than or equal to (Formula presented.). We provide many applications of our results, including the finiteness of homomorphisms from mapping class groups of nonorientable surfaces to (Formula presented.), the triviality of homomorphisms from the mapping class groups to (Formula presented.) or to (Formula presented.), and homomorphisms between mapping class groups. We also show that if the surface (Formula presented.) has (Formula presented.) marked point but no boundary components, then (Formula presented.) is generated by involutions if and only if (Formula presented.) and (Formula presented.).

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https://hdl.handle.net/11511/105101

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Department of Mathematics, Article