# Do discrete groups with property $(T)$ have “modest” subgroup growth?

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?

• $\mathrm{SL}_3(\mathbf{Z}[X])$ is known to have Property T.