I would like to gain some information about the discrete subgroups (lattices) of SU(N) Lie groups. I have already read some answers and references concerning the N=3 and N=4 cases. I am more interested in the large N limit.

More specifically I have two questions:

I would be very interested if someone knows about the existence of some kind of universality (a type of discrete group that appears very often for example) when considering the large N limit.

Taking into account that dimSU(N)= N^2-1, are there some peculiarities if we consider lattices with O(N^2) elements?