Abstract We present a practical algorithm which, given a non-archimedean local field K and any two elements A , B ∈ S L 2 ( K ) , determines after finitely many steps whether or not the subgroup 〈 A , B 〉 ≤ S L 2 ( K ) is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of S L 2 ( K ) by isometries on its Bruhat-Tits tree. The algorithm itself can also be used for two-generated subgroups of the isometry group of any locally finite simplicial tree, and has applications to the constructive membership problem. In an appendix joint with Frederic Paulin, we give an erratum to his 1989 paper ‘The Gromov topology on R -trees’, which details some translation length formulae that are fundamental to the algorithm.

中文翻译：

---

非阿基米德局部场上 SL2 的离散和自由二生成子群

摘要 我们提出了一个实用的算法，给定一个非阿基米德局部域 K 和任意两个元素 A , B ∈ SL 2 ( K ) ，在有限多步之后确定子群 〈 A , B 〉 ≤ SL 2 ( K ) 是离散的并且没有二阶。这利用了通过其 Bruhat-Tits 树上的等距图应用于 SL 2 ( K ) 动作的乒乓引理。该算法本身也可用于任何局部有限单纯树的等距群的两个生成子群，并应用于建设性隶属度问题。在与 Frederic Paulin 的附录中，我们给出了他 1989 年论文“R 树上的 Gromov 拓扑”的勘误表，其中详细介绍了一些对该算法至关重要的平移长度公式。