Let $\Gamma \leq \mathrm{Aut}(T_{d_1}) \times \mathrm{Aut}(T_{d_2})$ be a group acting freely and transitively on the product of two regular trees of degree $d_1$ and $d_2$. We develop an algorithm which computes the closure of the projection of $\Gamma$ on $\mathrm{Aut}(T_{d_t})$ under the hypothesis that $d_t \geq 6$ is even and that the local action of $\Gamma$ on $T_{d_t}$ contains $\mathrm{Alt}(d_t)$. We show that if $\Gamma$ is torsion-free and $d_1 = d_2 = 6$, exactly seven closed subgroups of $\mathrm{Aut}(T_6)$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\mathrm{Aut}(T_{6}) \times \mathrm{Aut}(T_{4n})$ and in $\mathrm{Aut}(T_{2n}) \times \mathrm{Aut}(T_{2n+1})$ respectively, for all $n \geq 2$. In particular we provide an explicit presentation of a torsion-free infinite simple group on $5$ generators and $10$ relations, that splits as an amalgamated free product of two copies of $F_3$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky-Mozes-Zimmer.

中文翻译：

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树及其投影的乘积中的新简单格子

令 $\Gamma \leq \mathrm{Aut}(T_{d_1}) \times \mathrm{Aut}(T_{d_2})$ 是一个群，对两个度数为 $d_1$ 的正则树的乘积自由传递和 $d_2$。我们开发了一种算法，该算法计算 $\mathrm{Aut}(T_{d_t})$ 上 $\Gamma$ 投影的闭包，假设 $d_t \geq 6$ 是偶数并且 $\ 的局部作用$T_{d_t}$ 上的 Gamma$ 包含 $\mathrm{Alt}(d_t)$。我们证明如果 $\Gamma$ 是无扭的并且 $d_1 = d_2 = 6$，$\mathrm{Aut}(T_6)$ 的七个闭合子群以这种方式出现。我们还在 $\mathrm{Aut}(T_{6}) \times \mathrm{Aut}(T_{4n})$ 和 $\mathrm{Aut}(T_{2n} }) \times \mathrm{Aut}(T_{2n+1})$ 分别，对于所有 $n \geq 2$。特别地，我们提供了一个关于 $5$ 生成器和 $10$ 关系的无扭无限单群的明确表示，它分裂为 $F_3$ 上 $F_{11}$ 的两个副本的合并自由乘积。我们包括从计算机辅助详尽搜索小度数树的产品中的格子所产生的信息。在 Pierre-Emmanuel Caprace 的附录中，我们的一些结果被用来表明自由群的抽象和相对公称群几乎是简单的，为 Lubotzky 和 ​​Lubotzky-Mozes-Zimmer 的问题提供了部分答案。