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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)?

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    $\begingroup$ I’m not sure of Serra’s motivation, but rank 1 uniform lattices are Gromov-hyperbolic groups. If a rank 1 uniform lattice has the congruence subgroup property, then there would be a hyperbolic groups which is not residually finite (in fact, one without any finite quotients). There are many Gromov hyperbolic groups known to be residually finite, but in general the question is wide open. $\endgroup$
    – Ian Agol
    Sep 22, 2022 at 5:07
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    $\begingroup$ Lubotzky recently gave a lecture where he discussed this issue and said that he believes that Serre's conjecture should fail for some Lie groups of rank 1, so the situation should be the same as property (T) and superrigidity. See the video and in particular the discussion around minutes 55-1:02. $\endgroup$
    – Amitay
    Sep 22, 2022 at 6:32
  • $\begingroup$ Similarly to the comment of @Amitay, in his second talk Lubotzky youtube.com/watch?v=pefDF9lkehQ mentions an observation (which he credits to Serre) that CSP implies superrigidity (For a given arithmetic lattice). He also mentions that if one were to prove the other direction, then Serre's conjecture would fail, in a way that would have Ian's answer as a consequence $\endgroup$
    – pitariver
    Sep 28, 2022 at 8:03
  • $\begingroup$ Time 22:35 in the above link $\endgroup$
    – pitariver
    Sep 28, 2022 at 8:09

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