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.