Larsen–Lubotzky conjecture on representation growth of irreducible lattices

Let $H$ be a higher-rank semisimple group of the form

$$H=\prod_i \mathbf{G}_i(F_i),$$

where each $F_i$ is a local field, each $\mathbf{G}_i$ is an absolutely almost simple $F_i$-group, and

$$\sum_i \operatorname{rank}_{F_i}(\mathbf{G}_i)\geq 2.$$

For a group $\Gamma$, let $\alpha_\Gamma$ denote the abscissa of convergence of its representation zeta function. Larsen–Lubotzky conjecture. For any two irreducible lattices $\Gamma_1,\Gamma_2\leq H$,

$$\alpha_{\Gamma_1}=\alpha_{\Gamma_2}.$$

This is a quantitative version of Serre's conjecture on the congruence subgroup property: the conjecture asserts that the abscissa of convergence is independent of the irreducible lattice in the higher-rank semisimple group. The paper proves the claim for the relevant groups of type $A_2$ in positive characteristic, assuming the congruence subgroup property for the irreducible lattices; the general conjecture remains open.