Varietes Riemanniennes Isospectrales et non Isometriques

Isospectral locally symmetric manifolds

“…To our knowledge the only other tower constructions are given by Vigérnas [30] for certain closed manifolds modeled on SL(2, R), SL(2, C), and products of these groups, and by Lubotzky-Samuel-Vishne [14] who constructed isospectral towers for certain pairs of closed locally symmetric manifolds modeled on SL(n, R) and SL(n, C) for n > 2. Even more remarkable is the fact that the towers constructed in [14] are for incommensurable manifolds M, N (see [8], [23], and [24]), an unobtainable feature via Sunada's method.…”

Isospectral manifolds with different local geometries

“…This was the first proof of the fact that the spectrum does not determine the isometry class of a Riemannian manifold. In 1980, M.-F. Vignéras discovered examples of isospectral Riemann surfaces and of isospectral hyperbolic manifolds in dimension three, the latter showing that the fundamental group is not spectrally determined ( [47]; see also P. Buser's book [9] on the spectral theory of Riemann surfaces). The first examples of continuous families of isospectral metrics were found by Carolyn Gordon and Edward Wilson in 1984 [25]; these were locally homogeneous metrics, induced by left invariant ones, on compact quotients of nilpotent or solvable Lie groups.…”

Traces, lengths, axes and commensurability

“…For closed hyperbolic manifolds, the only known constructions for producing isospectral manifolds are that of Sunada described in §4.1, and constructions building on one due to Vigneras [24]. In particular, all known constructions produce commensurable hyperbolic manifolds.…”