### Metadata

### Abstract

Let ${T}_{1},{T}_{2}$ be regular trees of degrees ${d}_{1},{d}_{2}\ge 3$. Let also $\Gamma \le \mathrm{Aut}\left({T}_{1}\right)\times \mathrm{Aut}\left({T}_{2}\right)$ 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 $\Gamma $ on ${T}_{i}$ is $2$-transitive; if moreover ${d}_{i}\ge 7$, assume that the local action contains $\mathrm{Alt}\left({d}_{i}\right)$. We show that $\Gamma $ is irreducible, unless $({d}_{1},{d}_{2})$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $\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 $\Gamma $ is irreducible, then it is hereditarily just-infinite, provided the local action on ${T}_{i}$ is not the affine group ${\mathbf{F}}_{5}\u22ca{\mathbf{F}}_{5}^{*}$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.

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