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

Caprace, Pierre-Emmanuel;Kassabov, Martin
(2023) Transactions of the American mathematical society — Vol. 376, n° 11, p. 7983-8021 (2023)

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Abstract
We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of Aut(F_p[x_1,…,x_n]) generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of Aut(F_p[x_1,…,x_n]) on the n-dimensional affine spaces over finite extensions of 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 p7−1 for any odd prime p.
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Caprace, P.-E., & Kassabov, M. (2023). Tame automorphism groups of polynomial rings with property (T) and infinitely many alternating group quotients. Transactions of the American mathematical society, 376(11), 7983-8021. https://doi.org/10.1090/tran/8988 (Original work published 2023)