A radius <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mn>1</mml:mn></mml:math> irreducibility criterion for lattices in products of trees

Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 \geq 3$. also $\Gamma \leq \mathrm{Aut}(T_1) \times \mathrm{Aut}(T_2)$ a group acting freely and transitively on $VT_1 VT_2$. For $i=1$ $2$, assume that the local action $\Gamma$ $T_i$ is $2$-transitive; if moreover $d_i 7$, contains $\mathrm{Alt}(d_i)$. We show irreducible, unless $(d_1, d_2)$ belongs to an explicit small set exceptional values. This yields irreducibility criterion for can checked purely in terms its ball radius $1$ $T_1$ $T_2$. Under same hypotheses, we then it hereditarily just-infinite, provided not affine $\mathbf F_5 \rtimes \mathbf F_5^*$. The proofs rely, several ways, Classification Finite Simple Groups.