# Examples of certain locally compact totally disconnected groups

To find a counterexample disproving a generalization of a theorem in the theory of scale functions on locally compact totally disconnected groups, initiated by George Willis, I am looking for a group with certain properties. Alternatively I am looking for arguments why such a group cannot exist. The precise question is:

Is there an example of a totally disconnected locally compact topological group $P$ such that
(1) $P$ is algebraically a (semi)direct product of subgroups $G$ and $H$
(2) $G$ and $H$ are not closed in $P$, but locally compact in the topology inherited from $P$,
(3) $K\cap G$ and $K\cap H$ are not compact for any compact open subgroup $K\subset P$,
(4) $P$ is not compact itself?

Usual construction methods of totally disconnected locally compact groups such as direct limits of discrete groups seem to fail to produce such an example. In particular (3) seems hard to achieve. So maybe an example which occurs naturally in some other context has a better chance to work than building one with bottom up methods.

I hope I didn't overlook any obvious argument contradicting the existence of the desired group.

• Condition (2) in itself seems extremely difficult to realize. I don't know of any example of pair of locally compact (group,subgroup) with that property. – André Henriques Sep 13 '10 at 16:34
• Dear Andre: Indeed, there are no examples if $P$ is meant to be Hausdorff (which surely it is). A locally compact subspace of a Hausdorff space is locally closed, hence open in its closure, so any such $G$ in open in its closure $\overline{G}$ in $P$, which in turn is a closed subgroup of $P$. But open subgroups of topological groups are closed, so $G$ is closed in $\overline{G}$ and hence $G = \overline{G}$. So (2) is impossible in the Hausdorff case, regardless of the other stuff. – BCnrd Sep 13 '10 at 18:09
• A tiny addition to BCnrd's comment (which seems fine to me): I found Bourbaki's book on topology really good for this sort of thing (i.e. definition of "locally closed" and so forth). – Matthew Daws Sep 13 '10 at 18:32
• Because every totally disconnected topological group is Hausdorff, BCnrd's answer is correct. Thank you very much! – Abel Stolz Sep 14 '10 at 8:17

First, note that every totally disconnected topological group is Hausdorff. Next any locally compact subspace of a Hausdorff space is locally closed, hence open in its closure (see e.g. Munkres Topology for a proof of this). If $$P$$ was a totally disconnected topological group and $$G$$ is a locally compact subgroup then $$G$$ is open in its closure $$\overline{G} \subset P$$, which in turn is a closed subgroup of P. But open subgroups of topological groups are closed, so $$G$$ is closed in $$\overline{G}$$ and hence $$G=\overline{G}$$. This proves (2) is impossible.