This is a question I don't believe is too vague to admit a sensible answer:
In what directions can the structure theory of the discrete subgroups of real and p-adic Lie groups applied? What situations might arise in which one could to delve into the theory of discrete groups to arrive at some other end?
Also,
Why might a number/representation theorist be interested in results like Ihara's which states that a torsion-free discrete subgroup of $\text{SL}_2(\mathbb{Q}_p)$ is free? (Aside from just the intrinsic interest of the result in a vacuum, which is still very interesting.)
Does the main interest lie in applying strong approximation to understand adelic quotients $G(\mathbb{A})/G(\mathbb{Q})$ (for nice algebaic groups $G$)?