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?
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6$\begingroup$ The $ax+b$ group of a $p$-adic field? $\endgroup$– Noam D. ElkiesJul 4, 2016 at 16:15
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3$\begingroup$ $S\ltimes \mathbb{R}$ for any subgroup $S<\mathbb{R}^*_+$? $\endgroup$– Uri BaderJul 4, 2016 at 18:01
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1$\begingroup$ For any countable group (since $G$ will have the topology of $|S|$ disjoint copies of ${\bf R}^*_+$). $\endgroup$– Noam D. ElkiesJul 5, 2016 at 12:42
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$\begingroup$ For any $S$ whatsoever, taking it with the discrete topology (and $\mathbb{R}$ with the usual one). In case $S$ is not countable, the group will not be second countable, though. This rases the question: which subgroups of $\mathbb{R}^*_+$ (or $\mathbb{R}$ to that matter) are homomorphic images of locally compact second countable groups. I suppose these are the countable ones, or $\mathbb{R}^*_+$ itself, and nothing else. But I haven't thought about it enough. $\endgroup$– Uri BaderJul 5, 2016 at 13:56
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1$\begingroup$ The $ax+b$ group over ${\bf R}$ is the standard example of a non-unimodular group, and is a semidirect product of the additive and multiplicative groups. @UriBader's suggestion is the subgroup of such transformations with $a \in S$. $\endgroup$– Noam D. ElkiesJul 5, 2016 at 14:54
1 Answer
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.