It is well-known that every compact Hausdorff group is the inverse limit of compact Lie groups (in fact, compact linear groups).
Question: is there any analogue for locally compact (or at least lcsc) groups? (This is a bit soft: I don't mean the expression to be neccessarilly as the inverse limit, or of Lie groups, but just some kind of decomposition into better-understood pieces.)
If I recall correctly, for compact groups, this follows for instance from the observation that the left regular Hilbert representation can be decomposed into finite-dimensional invariant subspaces.
This sort of argument does not appear to work at all for locally compact groups (at least to my untrained eye, with its rusty knowledge of representation theory). And I realise that this would include all discrete groups, so a simple inverse limit will probably not suffice. But maybe we can find some other kinds of constructions or "building blocks"?