A radius 1 irreducibility criterion for lattices in products of trees

Let $T_1,T_2$ be regular trees of degrees $d_1,d_2\ge 3$. Let also $\mathrm{\Gamma }\le \mathrm{A}\mathrm{u}\mathrm{t}(T_1)\times \mathrm{A}\mathrm{u}\mathrm{t}(T_2)$ be a group acting freely and transitively on $V{T}_1\times V{T}_2$. For $i=1$ and $2$, assume that the local action of $\mathrm{\Gamma }$ on $T_i$ is $2$-transitive; if moreover $d_i \geq 7$,assumethatthelocalactioncontains$$\mathrm{A}\mathrm{l}\mathrm{t}$(d_i)$\mathrm{.}Weshowthat$\Gamma$isirreducible,unless$(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $\mathrm{\Gamma }$ that can be checked purely in terms of its local action on a ball of radius~$1$ in $T_1$ and $T_2$. Under the same hypotheses, we show moreover that if $\mathrm{\Gamma }$ is irreducible, then it is hereditarily just-infinite, provided the local action on $T_i$ is not the affine group $\mathbf F_5 \rtimes \mathbf F_5^*$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.