What makes $p$-adic Lie groups interesting objects to study? I am not familiar with the representation theory or the number theory side of things, so I'd like to frame my question within the contexts of structural theories of topological groups and group action.
I know many theorems holds for Lie groups over local fields in general. For example, the superrigidity and arithmeticity theorems of Margulis, or the structural theorem of Nevo-Zimmer. However, it seems like many of the theorems are proven for real Lie groups first then generalized to the settings of p-adic Lie groups. I am looking for interesting structural theories unique to p-adic Lie groups that is not true on real Lie groups, or theories on real Lie groups that completely fails with no analogs expected in the case of p-adic Lie groups.
I know p-adic Lie groups are are totally disconnected. How does this topology come into play?