A radius 1 irreducibility criterion for lattices in products of trees

TL;DR

This paper establishes a criterion for irreducibility of certain lattice groups acting on products of trees, based on local actions, and shows these groups are hereditarily just-infinite under specific conditions.

Contribution

It provides a radius 1 irreducibility criterion for lattices in products of trees using local actions, with implications for their algebraic structure.

Findings

Irreducibility holds except for specific small degree pairs.

Irreducible groups are hereditarily just-infinite under certain conditions.

The criterion can be verified using local action data on radius 1 balls.

Abstract

Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 \geq 3$. Let also $\Gamma \leq \mathrm{Aut}(T_1) \times \mathrm{Aut}(T_2)$ be a group acting freely and transitively on $VT_1 \times VT_2$. For $i=1$ and $2$, assume that the local action of $\Gamma$ on $T_i$ is $2$-transitive; if moreover $d_i \geq 7$, assume that the local action contains $\mathrm{Alt}(d_i)$. 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 $\mathbf F_5 \rtimes \mathbf F_5^*$. The proof of irreducibility relies, in several ways, on the Classification of the Finite Simple Groups.