Complexity and randomness in the Heisenberg groups (and beyond)

By studying the commuting graphs of conjugacy classes of the sequence of Heisenberg groups $H_{2n+1}(p)$ and their limit $H_\infty(p)$ we find pseudo-random behavior (and the random graph in the limiting case). This makes a nice case study for transfer of information between finite and infinite objects. Some of this behavior transfers to the problem of understanding what makes understanding the character theory of the uni-upper-triangular group (mod p) "wild". Our investigations in this paper may be seen as a meditation on the question: is randomness simple or is it complicated?