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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?

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Why do you expect any peculiarities? One subgroup I know whose order is $O(N^2)$ is the Weyl-Heisenberg group generated by the two maps $|k\rangle\mapsto |k+1\rangle$ and $|k\rangle\mapsto e^{2\pi k/N}|k\rangle$, where $|k\rangle$, $k=0,\ldots, N-1 \pmod{N}$, is a orthonormal basis of $\mathbb C^N$. –  Yoav Kallus Jul 17 '12 at 13:57
    
Indeed I am expecting that, generically, the Cayley graph of a discrete subgroup will be an expander graph. But maybe this just happens for certain dimensions of the lattice. One way of rephrasing the question would be: take a discrete subgroup of SU(N) at random, Has it property T? Is it expanding? Do we need large N for this to happen? –  Felino Jul 19 '12 at 10:18
    
A larger finite subgroup than the Weyl-Heisenberg group is its normalizer, see e.g. arxiv.org/abs/1003.3591v2 . –  jjcale Aug 1 at 19:43

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