${\ell }^2$-stability and homomorphisms into the mapping class group

Abstract

We prove that any homomorphism into the mapping class group from a group satisfying a certain cohomological condition, called \textit{${\ell }^2$-stability}, is trivial. We show that such a condition is satisfied by all groups for which the above non-existence result was already known, hence providing in particular a unified proof in the above cases. We give moreover new examples of ${\ell }^2$-stable groups.