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

Understand

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.