Let $Q_p$ be the p-adic numbers, where p is any prime number. Then $Q_p$ is a locally compact, Hausdorff, totally disconnected (non-discrete) topological field. Let $GL(n,Q_p)$ be the general linear group over $Q_p$ equipped with subspace-product p-adic topology. So $GL(n,Q_p)$ is a locally compact, Hausdorff, totally disconnected (non-discrete) topological group. Does $GL(n,Q_p)$ contain a solvable cocompact subgroup which is closed in the p-adic topology?
Similarly, if $k$ is a finite extension of the p-adic numbers $Q_p$, does every linear algebraic group over $k$ have a solvable cocompact subgroup which is closed in the topology coming from $k$? (not in the Zariski topology)
Yes: $\mathrm{GL}(n,\mathbf{Q}_p)$ acts transitively on the variety of complete flags, which is compact, and the stabilizer of a point is the set of upper triangular matrices, which is solvable and cocompact. The same holds with $\mathbf{Q}_p$ replaced by any non-discrete locally compact field.