树的乘积格的半径1不可约准则
A radius 1 irreducibility criterion for lattices in products of trees
摘要：
设$T_1, T_2$是正则度树$d_1, d_2 \geq 3$。又设$Γ\leq \mathrm{Aut}(T_1) \times \mathrm{Aut}(T_2)$是自由传递作用于$VT_1 \times VT_2$的群。对于$i=1$和$2$，假设$Γ$在$T_i$上的局部动作是$2$ -可传递的;如果是$d_i \geq 7$，则假设局部动作包含$\mathrm{Alt}(d_i)$。我们表明，$Γ$是不可约的，除非$(d_1, d_2)$属于一个明确的小集合的例外值。这就产生了一个不可约的$Γ$准则，它可以纯粹根据它在$T_1$和$T_2$中对半径为~ $1$的球的局部作用来检验。在同样的假设下，我们证明了如果$Γ$是不可约的，那么它是遗传的正义无限的，只要$T_i$上的局部作用不是仿射群$\mathbf F_5 \rtimes \mathbf F_5^*$。不可约性的证明在几个方面依赖于有限单群的分类。
英文摘要：
Let $T_1, T_2$ be regular trees of degrees $d_1, d_2 \geq 3$. Let also $Γ\leq \mathrm{Aut}(T_1) \times \mathrm{Aut}(T_2)$ be a group acting freely and transitively on $VT_1 \times VT_2$. For $i=1$ and $2$, assume that the local action of $Γ$ 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 $Γ$ is irreducible, unless $(d_1, d_2)$ belongs to an explicit small set of exceptional values. This yields an irreducibility criterion for $Γ$ 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 $Γ$ 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.
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