Basic Non-Archimedean J{\o}rgensen Theory

We prove a non-archimedean analogue of J{\o}rgensen's inequality, and use it to deduce several algebraic convergence results. As an application we show that every dense subgroup of $\mathrm{SL}(\mathbb{Q}_p)$ contains two elements which generate a dense subgroup of $\mathrm{SL}(\mathbb{Q}_p)$, 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 $\mathrm{SL}(K)$ over a non-archimedean local field $K$ is discrete if and only if each of its two-generator subgroups is discrete.