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:
- What are these examples?
- 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?)
- What do we expect "modest growth" to mean?