Small isospectral and nonisometric orbifolds of dimension 2 and 3

Abstract

ABSTRACT. Revisiting a construction due to Vignéras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.Introduction. In 1966, Kac [54] famously posed the question: "Can one hear the shape of a drum?" In other words, if you know the frequencies at which a drum vibrates, can you determine its shape? Since this question was asked, hundreds of articles have been written on this general topic, and it remains a subject of considerable interest [45].Let (M, g) be a connected, compact Riemannian manifold (with or without boundary). Associated to M is the Laplace operator ∆, defined by) form an infinite, discrete sequence of nonnegative real numbers 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . , called the spectrum of M . In the case that M is a planar domain, the eigenvalues in the spectrum of M are essentially the frequencies produced by a drum shaped like M and fixed at its boundary. Two Riemannian manifolds are said to be Laplace isospectral if they have the same spectra. Inverse spectral geometry asks the extent to which the geometry and topology of M are determined by its spectrum. For example, volume, dimension and scalar curvature can all be shown to be spectral invariants.The problem of whether the isometry class itself is a spectral invariant received a considerable amount of attention beginning in the early 1960s. In 1964, Milnor [65] gave the first negative example, exhibiting a pair of Laplace isospectral and nonisometric 16-dimensional flat tori. In 1980, Vignéras [83] constructed non-positively curved, Laplace isospectral and nonisometric manifolds of dimension n for every n ≥ 2 (hyperbolic for n = 2, 3). The manifolds considered by Vignéras are locally symmetric spaces arising from arithmetic: they are quotients by discrete groups of isometries obtained from unit groups of orders in quaternion algebras over number fields. A few years later, Sunada [80] developed a general algebraic method for constructing Laplace isospectral manifolds arising from almost conjugate subgroups of a finite group of isometries acting on a manifold, inspired by the existence of number fields with the same zeta function where the group and its subgroups form what is known as a Gassmann triple [9]. Sunada's method is extremely versatile and, along with its variants, accounts for the majority of known constructions of Laplace isospectral non-isometric manifolds [30]. Indeed, in 1992, Gordon, Webb and Wolpert [50] used an extension of Sunada's method in order to answer Kac's question in the negative for planar surfaces: "One cannot hear the shape of a drum." For expositions of Sunada's method, the work it has influenced, and the many other constructions of Laplace isospectral manifolds, we refer the reader to surveys of Gordon [46,47] and the references therein.The examples of Vignéras are distinguished as they cannot arise from Sunada's method: in particular, they do not cover a common orbi...

Strongly isospectral hyperbolic 3-manifolds with nonisomorphic rational cohomology rings

“…It is then natural to ask: what is the smallest volume that an isospectral pair can have? Previous work of Linowitz and Voight [12] using number theoretic methods found a volume ≈ 51.02 example. The pair we give above has the smallest volume found so far, ≈ 25.418347, and we believe this would be difficult to improve significantly due to certain constraints.…”

On genera containing non-split Eichler orders over function fields

“…Also, BTB's play a significant role in the study of the selectivity problem, i.e., understanding when a commutative order embeds into all, or just into some, of the orders in a particular genus [12], [15], [16]. This problem arises naturally from questions regarding spectral properties of hyperbolic varieties [32], [17], and has been intensively studied in the quaternionic case, where the relevant buildings are trees.…”

Sunada transplantation and isogeny of intermediate Jacobians of compact Kähler manifolds

“…Sunada's technique was the first general technique for constructing isospectral manifolds and, due to its simplicity and power, it remains the most widely used method. For completeness, we note that an arithmetic method of constructing isospectral but nonisometric Riemann surfaces introduced by M.-F. Vignéras [38] and recently studied systematically by B. Linowitz and J. Voight [26] provides many examples of isospectral Riemann surfaces that do not arise from Sunada's construction. Outside of the setting of Riemann surfaces, other methods, including a general technique involving torus actions, have produced many interesting examples.…”