Basic Non-Archimedean Jørgensen Theory

We prove a non-archimedean analogue of Jørgensen’s inequality, and use it to deduce several algebraic convergence results. As an application we show that every dense subgroup of $\mathrm{S}\mathrm{L}({\mathbb{Q}}_p)$ contains two elements which generate a dense subgroup of $\mathrm{S}\mathrm{L}({\mathbb{Q}}_p)$, which is a special case of a result by Breuillard and Gelander. We also list several other related results, which are well-known to experts, but not easy to locate in the literature; for example, we show that a non-elementary subgroup of $\mathrm{S}\mathrm{L}(K)$ over a non-archimedean local field $K$ is discrete if and only if each of its two-generator subgroups is discrete.