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

We give a general method for proving Kazhdan’s property $(T)$ for quotients $G\mathrm{/}{K}_{[c+1]}$, where $G$ is countable, $K$ is a normal subgroup and $K_{[j]}$ denotes its lower central series. The method combines an affine realization of $G\mathrm{/}[K,K]$, a contraction argument, and permanence for nilpotent normal subgroups. We apply it to the Torelli lower-central quotients $\mathrm{M}\mathrm{o}\mathrm{d}({\mathrm{\Sigma }}_g)\mathrm{/}({\mathcal{T}}_g{)}_{[c+1]}$ for $g\ge 3$ and show that they have property $(T)$ for every $c\ge 1$. We also prove property $(T)$ for $\mathrm{A}\mathrm{u}\mathrm{t}(F_3)\mathrm{/}({\mathrm{I}\mathrm{A}}_3{)}_{[c+1]}$ for every $c\ge 1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. For the Torelli lower-central quotients with $g\ge 3$ and $c\ge 2$, every finite-dimensional complex representation has infinite kernel, and these quotients are not linear over any field.