Property $(T)$ and nonlinearity of mapping class group quotients

RESUMO

We give a general method for proving Kazhdan's property $(T)$ for quotients $G/K_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G/[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply it to the Torelli lower-central quotients $\mathrm{Mod}(\Sigma_g)/(\mathcal{T}_g)_{[c+1]}$ for $g\ge3$ and show that they have property $(T)$ for every $c\ge1$. We also prove property $(T)$ for $\mathrm{Aut}(F_3)/(\mathrm{IA}_3)_{[c+1]}$ for every $c\ge1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. For the Torelli lower-central quotients with $g\ge3$ and $c\ge2$, every finite-dimensional complex representation has infinite kernel, and these quotients are not linear over any field.