Representation zeta functions of groups of type $A_2$ in positive characteristic

TL;DR

This paper proves two conjectures related to the representation growth of groups of type A_2, focusing on uniformity of zeta functions over local rings and growth in lattices in positive characteristic.

Contribution

It confirms the uniformity conjecture for representation zeta functions and the Larsen--Lubotzky conjecture for groups of type A_2 in positive characteristic.

Findings

Proves the uniformity of representation zeta functions over local rings.

Establishes the growth behavior of irreducible lattices in groups of type A_2.

Supports the Larsen--Lubotzky conjecture assuming Serre's conjecture.

Abstract

We prove two conjectures regarding the representation growth of groups of type $A_2$. The first, conjectured by Avni, Klopsch, Onn and Voll, regards the uniformity of representation zeta functions over local complete discrete valuation rings. The second is the Larsen--Lubotzky conjecture on the representation growth of irreducible lattices in groups of type $A_2$ in positive characteristic assuming Serre's conjecture on the congruence subgroup problem.