It is easy to find examples of locally compact second countable Hausdorff topological groups $G$ whose modular function $\Delta$ has image $\{1\}$ or $(0,\infty)$. Are there groups $G$ of this kind for which the image of $\Delta$ is *anything* else?



Haar measure, can image of modular function be any subgroup of $(0,\infty)$?
I am not absolutely sure what is the question.
The answer to the question appearingin in the body is given in a comment by Noam Elkies and the answer to the question given in the title is given by a comment of mine.
Let me answer a third question which is implied and it is less trivial:
which subgroups of $\mathbb{R}^\*\_+$ appear as images of the modular homomorphism of second contable locally compact groups (lcsc).


I claim the list is $\{1\}$, $\mathbb{R}^\*\_+$ and all of its countable subgroups.


The class of subgroups under consideration coincides with the class of subgroups of $\mathbb{R}^\*\_+$ obtained as all images of continuous homomrphism of lcsc groups, as could be seen by a semi direct product construction. In fact, I may consider only injective homorphisms, by moding out the kernel. In particular, I may assume my groups are abelian and with no compact subgroups. The connected component of such group is (isomorphic to) $\mathbb{R}^n$, hence must be mapped onto for $n>0$ (and in this case $n=1$), so I may assume my group is totally disconnected. A totally disconnected group with no nontrivial compact subgroup must be discrete, so my group is actually countable.


