Division Algebras and Isospectral Manifold

Two compact Riemanian manifolds are said to be isospectral if
the (multi-set of ) the eigenvalues of their Laplacians are equal. The
question of finding isospectral non-isometric manifolds has a long
history starting with Marc Kac’s seminal paper (1966): ” Can you hear the
shape of a drum?”
Various methods are known to construct such manifolds- the
most powerful is due to Sunada. For locally symetric manifolds $M$
(i.e., $M= D\G/K , G$ semisimple Lie groups, $K$ maximal compact subgroup and
$D$ a lattice in $G$) all the know examples are commensurable to each other.
In fact, Reid showed that if $G=SL(2,R$ and $D$ arithemetic these are the
only possibilities. We show that for $G=SL(n,R), n>2$, the situation is
different and there are non-commensurable examples. The different between $n=2$ and $n>2$ lies in the fact that division algebras of degree $2$ (i.e.,
quaternion algebras) are determined by their ramification primes while
this is not true for $n>2$. We will also present some related isospectral Cayley
graphs of some finite simple groups.(Joint with B. Samuels and U. Vishne.)