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

Abstract

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 this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients $\mathrm{Mod}({\Sigma }_g)/({\mathcal{T}}_g{)}_{[c+1]}$ for $g\ge 3$ we show that they have property $(T)$ for every $c\ge 1$. We also prove property $(T)$ for $\mathrm{Aut}(F_3)/({\mathrm{IA}}_3{)}_{[c+1]}$ for every $c\ge 1$, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for $c\ge 2$ are not residually finite, and thus non-linear.

arXiv commentTwo notable improvements: 1) added more examples of groups to which the main criterion for property (T) applies; 2) improved non-linearity of the Torelli lower central series quotients to non-residual finiteness. Some minor editing and formatting. 35 pages