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

2022/10/03 by Pierre‐Emmanuel Caprace, Caprace, Pierre-Emmanuel, Martin Kassabov +1
Mathematics · #05C48 #14E07 #20D06 #20F67 #22D55 #Advanced Differential Equations and Dynamical Systems #Commutative Algebra (math.AC) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2210.00730

openalex publication_date 2022/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct new families of groups with property (T) and infinitely many alternating group quotients. One of those consists of subgroups of Aut(\mathbf F_p[x₁, …, x_n]) generated by a suitable set of tame automorphisms. Finite quotients are constructed using the natural action of Aut(\mathbf F_p[x₁, …, 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 p⁷-1 for any odd prime p.

Related

• A new co-tame automorphism of the polynomial ring
• Generalisations of the tame automorphisms over a domain of positive characteristic
• Aut-invariant quasimorphisms on groups
• Polyadic groups and automorphisms of cyclic extensions
• Property (T) and Many Quotients