How can one approximate compact Lie groups by finite groups? My wish is something like this:
Let $G$ be a compact Lie group. There is a sequence of nested finite subgroups $G_n$ so that $G_n\to G$ in the Hausdorff distance. (That is, for every $\epsilon>0$ there is $N$ so that the $\epsilon$-neighbourhood of $G_n$ is $G$ for all $n\geq N$.)
This is true if $G$ is abelian or if we drop all group structure from the statement (working with compact manifolds and finite subsets). I found a paper by Turing stating that this indeed fails for all nonabelian connected Lie groups. Are there any positive results in the same spirit for nonabelian groups?
For example, could there be a sequence of finite groups $G_n$ (with suitable metric) which are not necessarily subgroups of $G$ so that $G_n\to G$ in the Gromov-Hausdorff sense? The exact mode of convergence is not important; I would just like to know if this idea has any hope in general (or for any single nonabelian group).
The best I could come up with is the following: Let $G$ be a nonabelian, compact, connected Lie group. Let $\epsilon>0$. Given $g\in G$, let $T<G$ be a maximal torus containing $g$ and find a finite subgroup $H_g<T$ so that its $\epsilon$ neighborhood covers $T$. By compactness $\{H_g;g\in A\}$ covers $G$ for some finite $A\subset G$. This gives the following result, which is weaker than my wish:
Let $G$ be a compact Lie group. There is a nested sequence $(C_n)$ of finite collections of finite abelian subgroups of $G$ so that $U_n\to G$ in the Hausdorff distance, where $U_n=\bigcup_{H\in C_n}H$ is the union of the finite groups in $C_n$.
In a different direction, I know that every finite group can be embedded in a compact, connected Lie group, but that doesn't seem to be of much help.