Suppose one has a connected compact Lie group $K$, and an element $k\in K$ of infinite order. Then can the subgroup generated by $k$ be discrete?
I suspect the answer is no, but this suspicion is mostly based on the example of irrational rotations in $S^1$, where the powers of an irrational rotation have accumulation points and hence cannot form a discrete subgroup. But can this argument be adapted to get the result for a general compact connected Lie group?