1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 2. RAMANUJAN HYPERGRAPHS 417

of (7) vectors w with jSupp(w)I = i. Partition the representatives in S i according

to the type w E h For each w, there are

(2.1)

representatives of type w; choose them to have the following form: its jj-th entry

is 7f if and only if j E Supp(w). For f > j, its jf-th entry is 0 unless its jj-th entry

is 7f and its ff-th entry is 1, in which case the jf-th entry lies in Op modulo 7r0p.

Call these basic matrices of type w.

Exercise. Show that

qn,i = L qe(w).

wEli

This then verifies the regularity condition (R).

The commutativity condition (C) can be easily checked by verifying (U). There-

fore Bn,F is a (q + 1)-regular n-hypergraph, and every vertex in Bn,F can be repre-

sented by a finite product of basic matrices of types in LJ~/ h Since Bn,F is topo-

logically contractible, it is the universa l cover of all (q + 1)-regular n-hypergraphs.

This is the higher dimensional analogue of the tree PGL 2 (F)/PGL 2 (0F ).

The operator spectrum of Ai acting on the space of L^2 -functions on Bn,F is

well-known. See for example Macdonald's books [26], [27] for more detail. Denote

by O'i(z 1 , ... , Zn) the i-th elementary symmetric polynomial in z 1 , ... , Zn, that is ,

O'i(Z1, ... , Zn) = L z;,"' · · · z~n.

w=(w,, ... ,wn)El;

Let

Dn,i = { O'i (z1, ... , Zn) : Z1,... , Zn E C x , lz1 I = · · · = lznl = 1, Z1 · · ·Zn = 1 }.

Then the spectrum of Ai for 1 ~ i ~ n - i is dilation by qi(n-i)/^2 of Dn,i· Clearly
Ai and An-i have the same spectrum. The geometric shape of Dn,i was explicitly
described by Cartwright and Steger [4]; they showed that the boundary of Dn,i is
the curve traced by O' i ( e v'=I^11 , ••• , e v'=I^11 , e-v'=I ( n-l )O) as () moves from 0 to 2 7f.

2.2. The growth of the spectra of finite quotients of the Bruhat-

Tits building

The goal of this section is to establish a higher dimensional analogue of the Alon-
Boppana theorem, namely, given a family of finite (q + 1)-regular n-hypergraphs
{ Xj} with IXj I ---; oo as j ---; oo, for each 1 ~ i ~ n - 1, the closure of the set of
nontrivial eigenvalues of Ai on Xj for all j contains the spectrum of Ai on Bn,F.
As we have learned from studying lim sup,\ - of graphs, this would not be true
unconditionally. The condition we impose on Xj is to require the radius of Xj to
go to infinity as j ---; oo. Here a (q + 1)-regular n-hypergraph X is said to have
ragi:us d if d is the largest integer m such that there is a geodesic ball Bu ( m) in
X centered at some vertex u in X with radius m which is isomorphic to a ball of
radius min the building Bn,F·