Tame automorphism groups of polynomial rings with property (T) and infinitely many alternating group quotients

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of $\mathrm{A}\mathrm{u}\mathrm{t}({\mathbf{F}}_p[x_1,\dots ,x_n])$ generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of $\mathrm{A}\mathrm{u}\mathrm{t}({\mathbf{F}}_p[x_1,\dots ,x_n])$ on the $n$-dimensional affine spaces over finite extensions of ${\mathbf{F}}_p$. As a consequence, we obtain explicit presentations of Gromov hyperbolic groups with property (T) and infinitely many alternating group quotients. Our construction also yields an explicit infinite family of expander Cayley graphs of degree $4$ for alternating groups of degree $p^7-1$ for any odd prime $p$.