  We construct new families of groups with property (T) and infinitely many
alternating group quotients. One of those consists of subgroups of
$\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ generated by a suitable set of
tame automorphisms. Finite quotients are constructed using the natural action
of $\mathrm{Aut}(\mathbf F_{p}[x_1, \dots, x_n])$ 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 $p$.
