\(\ell 2\)-stability and homomorphisms into the mapping class group (Q6150773)

Summary: We prove that any homomorphism into the mapping class group from a group satisfying a certain cohomological condition, called \(\ell^2\)-\textit{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. For the entire collection see [Zbl 1519.57002].