The old question beautifully formulated by Mark Kac as “Can one hear the shape of a drum?” has been of interest over the years both to geometers as well as to Lie group theorists. In the 1980’s, Sunada came up with a novel method in analogy with the way one produces non-isomorphic algebraic number fields with the same Dedekind zeta function. His simple method led to several examples of isospectral, but not isometric, Riemann surfaces. Nevertheless, several questions remained unanswered. For instance, in this paper, it is shown for the first time that the symmetric space \(X\) corresponding to any non-compact simple Lie group, covers two closed, isospectral, non-isometric manifolds. In fact, the author shows the existence of two closed manifolds \(M,N\) covered by \(X\) which have infinite towers \(\{M_j \}_j\) and \(\{N_j \}_j\) of finite covers such that \(M_j\) and \(N_j\) are isospectral but not isometric. More precisely, the author proves:
Let \(G\) be a non-compact simple Lie group and \(X\) its associates symmetric space. Then, for every positive integer \(n\), there exist \(n\) closed, isospectral, non-isometric manifolds whose universal cover is \(X\).
Even for the much-studied case of hyperbolic manifolds, the above theorem provides many new examples. For simple Lie groups of exceptional type, the result is entirely new. To describe the other results of the paper, denote by \(SD_X(t)\) the cardinality of the largest set of isometry classes of closed manifolds that are covered by \(X\), are pairwise isospectral and have volumes at the most \(t\). This is a finite number for each positive real number \(t\). These numbers were previously not known to be unbounded. Here, the author proves:
For each positive integer \(r\), there exists a strictly increasing sequence \(\{t_j\}_j\) of positive real numbers such that \(SD_X(t_j) \geq t_j^r\).
When \(X\) is a hyperbolic space, the author provides a finer lower bound:
Let \(X\) be either a real hyperbolic space or the complex hyperbolic \(2\)-space. Then, there exists a constant \(D\) and a strictly increasing sequence \(\{t_j \}_j\) of positive real numbers such that \(SD_X(t_j) \geq t_j^{D \log(t_j)}\).
The main idea used in the proofs is Sunada’s method applied to certain subgroups of \(3\)-dimensional Heisenberg groups over finite fields. Strong approximation allows and assures the existence of homomorphisms surjecting onto such finite groups. To ensure the existence of non-isometric covers the author employs Chebotarev’s density theorem.
Clearly, the author has a gift for communication and the paper is interspersed with several useful remarks which guides the reader through the pitfalls as well as high points of the subject matter.