352 AUDREY TERRAS, ARITHMETICAL QUANTUM CHAOS
spectrum for finite upper half plane over ring Zl169Z, d=2,a=17
10080604020(^0) --- --
-30 -20 -10^10 20 30
level spacing for finite upper ha~ plane over ring Zl169Z, d=2,a=17
150
100
50(^0) -
0 0.5 1.5 2.5 3.5
Figure 1 0. Histograms of Spectra (without multiplicity) of Finite Upper Half
P lane Graphs over the Ring Z/169Z with d = 2 and a = 17. The top histogram
is that for the spectrum and the bottom one is that for the unnormalized level
spacing. Data was computed by B. Shook using Matlab. See Angel et al [4 ].
4.3. Dream s o f " Larger" Gro u p s and Butterflies.
What happens if you consider "larger" matrix groups such as GL(3, F)? What
happens if you replace graphs with hypergraphs or buildings? Do new phenomena
appear in the spectra? See Fan Chung's article in Friedman [31] as well as K. Feng
and Winnie Li [28] and Li and P. Sole [5 1] for hypergraphs. In [56] Maria Martinez
constructs some hypergraphs analogues of the symmetric space for G L( n, IF q) using
n-point invariants and shows that some of these hypergraphs are Ramanujan in the
sense of Feng and Li. See also [5 7]. Recently Cristina Ballantine [5] and Winnie
Li [50] have found an analogue of the Lubotzky, Phillips and Sarnak examples
of Ramanujan graphs for GL(n) using the theory of buildings, which are higher
dimensional analogues of trees. See also the Ph.D. thesis of Alireza Sarveniazi at
the Universitat Gottingen. Nancy Allen [2] and F. J. Marquez [55] have considered
Cayley graphs for 3x3-matrix analogues of the affine group of matrices ( ~ ~ ) ,
with distances analogous to that for the finite upper half plane graphs.
An important subgroup of GL(3) corresponding to the s-coordinates in the fi-
nite upper half plane is the H e isenberg group H(F) over a ring or field F. H(F)