Complexity and randomness in the Heisenberg groups (and beyond)

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