# When can a locally compact group be approximated by discrete subgroups?

This question is about partitioning a (locally) compact group into cells by using discrete subgroups.

Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a metrizable group.) Say that $G$ is “approximable by discrete subgroups” if there is a sequence of countable/finite locally compact subgroups $G_1 \subseteq G_2 \subseteq G_3 \subseteq ... \subseteq G$ such that for all $n$, $G/G_n$ is a compact group and $\bigcup_n G_n$ is dense in $G$.

This property clearly holds for $(\mathbb{R}^n,+)$, addition on the torus, and products of finite groups.

• Question 1: Is there a characterization of locally compact groups “approximable by discrete subgroups”? For example, is every metrizable locally compact group “approximable by discrete subgroups”? Every Lie group?

My motivation is that such groups can be partitioned into cells based on these approximations. This allows me to discretize the group and solve the problem I am currently working on. (I also need to apply other methods to handle points which are close together but in different cells.) I am basically trying to generalize a proof for addition on $\mathbb{R}^n$ to other locally compact metrizable groups.

• Question 2: (Vague!) If not all metrizable compact groups are “approximable by discrete subgroups,” is there any well-known techniques to nicely divide up a compact group into regions which are almost translates of each other, or something similar to the above idea.

Disclaimer: I apologize if this is not at the level of MO. It is a research question, but I know very little about topological groups. Also, I am happy to get pointed to useful resources on this subject.

• Are you assuming in your definition that each $G_n$ is closed in $G$? Without this assumption, the quotient topology on $G/G_n$ won't be Hausdorff, I think – Yemon Choi Jan 6 '16 at 23:37
• I fear that this property (strictly required) is difficult to achieve... My guess would be that, among real Lie groups, for example, only nilpotent groups would have this property. But this is just a superficial reaction... – paul garrett Jan 6 '16 at 23:37
• A countable (= at most countable) locally compact group is discrete. So the weird "locally compact subgroup" just means discrete subgroup, and a discrete subgroup $H$ such that $G/H$ is compact is known as a "cocompact lattice" or "uniform lattice". – YCor Jan 6 '16 at 23:48
• It is an old result of H. Toyama that if a connected Lie group is approximable by discrete subgroups, then it is nilpotent (and this even holds assuming only that the subgroups are discrete, not necessarily cocompact). By the way your definition should at least be called "approximable by discrete cocompact subgroups". – YCor Jan 6 '16 at 23:51
• Btw I should add the usual definition of approximable by discrete subgroups (for a 2nd countable locally compact group $G$): the existence of a sequence $(H_n)$ of discrete subgroups such that for every nonempty open subset $U$ of $G$ there exists $N=N_U$ such that $H_n\cap U\neq\emptyset$ for all $n\ge N$ (if $(H_n)$ is ascending this just means that the union is dense). The above characterizations anyway are correct with either definition. – YCor Jan 7 '16 at 1:14