I saw it conjectured at http://www.mathunion.org/ICM/ICM1994.1/Main/icm1994.1.0309.0317.ocr.pdf that "discrete subgroups with property $(T)$ may have modest subgroup growth." (Page 5, directly above the heading for section 4.)

According to the author, this conjecture was supported by some examples.

I have three questions:

  1. What are these examples?
  2. Has any progress been made on this conjecture - are there nontrivial bounds on the subgroup growth of a discrete group with property $(T)$? (What are the trivial bounds?)
  3. What do we expect "modest growth" to mean?

This is now outdated: in 1994 there were very few known sources of Property T groups: lattices on the one hand (for which congruence subgroup property was true or unknown, and hyperbolic groups with no information on their finite index subgroups). This has dramatically changed. Restricting to cases for which we have information about finite quotients, we have, for instance

  • $\mathrm{SL}_3(\mathbf{Z}[X])$ is known to have Property T.
  • Also there are Golod-Shararevich groups with Property T (Ershov, see here ). See Jaikin-Zapirain's appendix to Ershov-Jaikin's paper showing that the subgroup of Golod-Shafarevich groups is large.

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