There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$.
- Whether a lattice in the group satisfies the congruence subgroup property,
- Kazhdan's property (T),
- Local rigidity/superrigidity,
- Bounded generation.
Serre conjectured that the congruence subgroup property should fail for a simple, connected Lie group of real rank 1, or more generally for a simple algebraic group over a local field $K$ of $K$-rank 1. On the other hand, the other properties like superrigidity or Kazhdan's property (T) seem to have a similar dichotomy between rank 1 and rank $\ge2$, although the boundary cases are slightly different; for example, the Lie group $\mathrm{Sp}(n,1)$ has property (T) and superrigidity, although it has real rank 1.
As an outsider, I wonder why Serre's conjecture has such a clean description while the other related properties do not. What is the fundamental difference between rank 1 and rank $\ge2$ that makes this dichotomy appear, and why does this work for congruence subgroup property and not for the related properties like Kazhdan's property (T)?