Triviality of some representations of $\operatorname {MCG}(S_g)$ in $GL(n,\mathbb {C}), \operatorname {Diff}(S^2)$ and $\operatorname {Homeo}(\mathbb {T}^2)$

Abstract:

We show the triviality of representations of the mapping class group of a genus $g$ surface in $GL(n,\mathbb {C}), \operatorname {Diff}(S^2)$ and $\operatorname {Homeo}(\mathbb {T}^2)$ when appropriate restrictions on the genus $g$ and the size of $n$ hold. For example, if $S$ is a surface of finite type with genus $g \ge 3$ and $\phi : \operatorname {MCG}(S) \to GL(n,\mathbb {C})$ is a homomorphism, then $\phi$ is trivial provided $n < 2g.$ We also show that if $S$ is a closed surface with genus $g \ge 7$, then every homomorphism $\phi : \operatorname {MCG}(S) \to \operatorname {Diff}(S^2)$ is trivial and that if $g \ge 3$, then every homomorphism $\phi : \operatorname {MCG}(S)\to \operatorname {Homeo}(\mathbb {T}^2)$ is trivial.