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

TL;DR

This paper constructs new groups with property (T) and infinitely many alternating group quotients, including explicit hyperbolic groups and expanders, by leveraging tame automorphisms of polynomial rings over finite fields.

Contribution

It introduces a novel method to generate groups with property (T) and many alternating quotients using tame automorphisms, expanding the understanding of such groups.

Findings

Constructed groups with property (T) and infinitely many alternating group quotients.

Provided explicit presentations of Gromov hyperbolic groups with these properties.

Produced explicit families of expander Cayley graphs for large alternating groups.

Abstract

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 $p$.