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

TL;DR

This paper introduces a practical algorithm to determine whether a subgroup generated by two elements in SL_2 over a non-archimedean local field is discrete and free, using the Bruhat-Tits tree and the Ping Pong Lemma.

Contribution

The paper presents a new algorithm for identifying discrete free subgroups of SL_2 over non-archimedean fields, extending to actions on locally finite trees and addressing the constructive membership problem.

Findings

Algorithm successfully determines subgroup discreteness and freeness.

Applicable to isometry groups of locally finite trees.

Includes an erratum with translation length formulas for the Gromov topology.

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.