Discrete and free two-generated subgroups of ${\rm SL_2}$ over non-archimedean local fields

Abstract

We present a practical algorithm which, given a non-archimedean local field $K$ and any two elements $A,B\in {\rm SL_2}(K)$, determines after finitely many steps whether or not the subgroup $\langle A, B \rangle\le {\rm SL_2}(K)$ is discrete and free of rank two. This makes use of the Ping Pong Lemma applied to the action of ${\rm SL_2}(K)$ 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 Fr\'ed\'eric Paulin, we give an erratum to his 1989 paper `The Gromov topology on $\mathbb{R}$-trees', which details some translation length formulae that are fundamental to the algorithm.