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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)

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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.

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  • $\begingroup$ Dear Yves, Thanks. It is surprising that it is true for ALL local fields. By the way, do you know any books/papers about structure of linear algebraic group over non-archimedean local fields of characteristic p (for p any given prime number)? $\endgroup$ – m07kl Apr 18 '14 at 9:20
  • $\begingroup$ No... I don't even know a reference for the following (very likely) fact: for any non-discrete locally compact field $K$ and every algebraic $K$-group $G$, the group $G(K)$ has a cocompact solvable subgroup. $\endgroup$ – YCor Apr 18 '14 at 10:20
  • $\begingroup$ If characteristic is zero look at Proposition 9.3 in "A. Borel and J. Tits: Groupes reductifs" For GLn case look at page 13 Lemma 5 in "GUENTNER, HIGSON and WEINBERGER: THE NOVIKOV CONJECTURE FOR LINEAR GROUPS" $\endgroup$ – m07kl Apr 18 '14 at 10:57
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    $\begingroup$ I don't claim they proved it, I claimed they knew it. One way you can do is to write something like: "it is well-known that the upper triangular group is cocompact in $GL(n,F)$ (for instance, because it is the stabilizer of a point in the transitive action on the variety of complete flags)". $\endgroup$ – YCor Apr 21 '14 at 19:23
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    $\begingroup$ I claimed this only when $K$ has characteristic zero (although I believe it works in general). Basically I use a minimal parabolic in the reductive case, and in general I get one modulo the unipotent radical and pull it back. The cost is a few minor technicalities I'm lazy to develop right now. An alternative argument is to show that, using the adjoint representation, there is a closed orbit in the flag variety; then point stabilizers are cocompact solvable subgroups. $\endgroup$ – YCor Apr 21 '14 at 19:48

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