On the linearity problem for mapping class groups

Abstract

Formanek and Procesi have demonstrated that $Aut\left(F_n\right)$ is not linear for $n\ge 3$. Their technique is to construct nonlinear groups of a special form, which we call FP-groups, and then to embed a special type of automorphism group, which we call a poison group, in $Aut\left(F_n\right)$, from which they build an FP-group. We first prove that poison groups cannot be embedded in certain mapping class groups. We then show that no FP-groups of any form can be embedded in mapping class groups. Thus the methods of Formanek and Procesi fail in the case of mapping class groups, providing strong evidence that mapping class groups may in fact be linear.