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.