In "Sunada's method and the covering spectrum" (J. Differential Geom. 86 (2010), 501-537), de Smit, Gornet and Sutton extended Sunada's method — the standard technique for producing Riemannian manifolds with identical Laplace spectra — to the Sormani-Wei covering spectrum, the part of the weak length spectrum that measures the size of a manifold's one-dimensional holes. They proved that a purely group-theoretic jump condition on a finite group GG with subgroups HH and H′H' is equivalent to the quotient manifolds H\MH\backslash M and H′\MH'\backslash M having equal covering spectra for every GG-invariant metric whenever the covering manifold MM is simply connected and of dimension at least 3; in general, the jump condition implies covering-spectral equality for every simply connected MM in any dimension, and in dimension at least 3 the converse, that equal covering spectra for all invariant metrics forces the jump condition, holds without simple connectivity. The reverse implication in dimension two is explicitly left open: "We do not know if (4) implies (3) for all 2-dimensional manifolds." The open surface case is whether equality of covering spectra for all invariant metrics forces the jump condition on surfaces, or whether there exists a non-jump triple whose surface quotients nevertheless have equal covering spectra for every invariant metric. Follow-up literature has resolved adjacent questions -- the covering spectrum is not a Laplace spectral invariant, in every dimension at least 2 -- but the dimensional gap in the equivalence theorem remains.

Background

Two central invariants of a closed connected Riemannian manifold (M,g)(M,g) are the Laplace spectrum, i.e. the sequence of eigenvalues of the Laplace operator acting on smooth functions, and the length spectrum, the multiset of lengths of smoothly closed geodesics, each length counted with multiplicity equal to the number of free homotopy classes (deformation classes of unbased loops) containing a geodesic of that length. Ignoring multiplicities gives the weak (or absolute) length spectrum. How these two invariants constrain one another is a classical inverse-spectral theme: Colin de Verdiere proved that for a generic metric (one whose closed geodesics are all nondegenerate, often called bumpy) the weak length spectrum is determined by the Laplace spectrum, and Duistermaat and Guillemin showed that for an arbitrary metric the Laplace spectrum at least determines the singular support of the wave trace, a subset of the weak length spectrum; whether the weak length spectrum is a spectral invariant in general remains open.

In 2004, Sormani and Wei introduced the covering spectrum, a geometric invariant that isolates a finite part of the weak length spectrum and, roughly, measures the size of the one-dimensional holes of the space. For a compact Riemannian manifold, one assigns to each non-trivial covering N→MN\to M the number r(N/M)r(N/M), defined as half the length of the shortest closed geodesic in MM that has a lift to NN which is not a closed loop; the covering spectrum CovSpec(M,g)\mathrm{CovSpec}(M,g) is the set of these values, so 2 CovSpec(M,g)2\,\mathrm{CovSpec}(M,g) is a subset of the weak length spectrum. An equivalent formulation used in the source paper goes through the minimum marked length map mg:π1(M)→R≥0m_g:\pi_1(M)\to\mathbb{R}_{\ge 0}, which assigns to each element of the fundamental group the length of the shortest closed geodesic in its free homotopy class. This map is a length map in an algebraic sense: it is positive away from the identity, constant on conjugacy classes, and satisfies m(gk)≤∣k∣m(g)m(g^k)\le|k|m(g). For δ>0\delta>0 the elements with mg(g)<δm_g(g)<\delta generate a normal subgroup Filδ(mg)\mathrm{Fil}_\delta(m_g), giving an increasing filtration of π1(M)\pi_1(M) indexed by the positive reals, and the covering spectrum is exactly half the jump set of this filtration, i.e. the set of δ\delta at which the filtration changes. The covering space of MM associated with the subgroup Filδ(mg)\mathrm{Fil}_\delta(m_g) is called the δ\delta-cover of MM. Sormani and Wei observed that certain Sunada-isospectral manifolds share the same covering spectrum, which raised the question of whether the covering spectrum is audible, that is, a spectral invariant.

Sunada's method, the most widely used technique for constructing isospectral manifolds (Riemannian manifolds with the same Laplace spectrum), starts from a finite group GG with subgroups HH and H′H' such that #(H∩C)=#(H′∩C)\#(H\cap C)=\#(H'\cap C) for every conjugacy class CC of GG; such a triple (G,H,H′)(G,H,H') is called a Gassmann-Sunada triple, and HH and H′H' are said to be almost conjugate or Gassmann-Sunada equivalent. If GG acts freely on a closed connected manifold MM, Sunada's theorem gives that the quotients H\MH\backslash M and H′\MH'\backslash M are isospectral for every GG-invariant Riemannian metric, and the converse implication (isospectrality for all such metrics forces the algebraic condition) also holds, so the Gassmann-Sunada condition is exactly the criterion for Sunada's construction to preserve the Laplace spectrum.

de Smit, Gornet and Sutton established the covering-spectrum analogue. They define (G,H,H′)(G,H,H') to be a jump triple if for all subsets S,T⊆GS,T\subseteq G that are stable under conjugation, ⟨H∩S⟩=⟨H∩T⟩  ⟺  ⟨H′∩S⟩=⟨H′∩T⟩,\langle H\cap S\rangle=\langle H\cap T\rangle \iff \langle H'\cap S\rangle=\langle H'\cap T\rangle, where ⟨⋅⟩\langle\cdot\rangle denotes the subgroup generated by a subset; equivalently, every length map on GG restricts to HH and to H′H' with identical jump sets. Their Theorem 2.2 states: if MM is simply connected, then the jump condition implies that H\MH\backslash M and H′\MH'\backslash M have the same covering spectrum for every GG-invariant metric; and if MM has dimension at least 3, then conversely, equality of covering spectra for every GG-invariant metric forces (G,H,H′)(G,H,H') to be a jump triple (this direction needs no simple connectivity). For simply connected MM of dimension at least three, the jump condition is therefore exactly the criterion for when Sunada's construction preserves the covering spectrum. A byproduct is that Gassmann-Sunada triples failing the jump condition exist -- built from the wreath product V⋊GL(V)V\rtimes\mathrm{GL}(V) of a finite vector space by its general linear group and from a variant of Todd's 1949 construction in symmetric groups -- the smallest of index [G:H]=7[G:H]=7; this yields Sunada-isospectral manifolds with distinct covering spectra in every dimension at least 3, and hence the covering spectrum is not a spectral invariant there.

The proof of the implication (covering-spectral equality for all invariant metrics implies jump equivalence) in dimensions at least 3 rests on a prescription theorem: on a closed manifold of dimension at least 3, given finitely many homotopy classes and admissible target lengths, one can construct a Riemannian metric whose minimum marked length map takes exactly those values on those classes and is uniformly large elsewhere. This works because in dimension at least 3 any finite collection of free homotopy classes can be represented by pairwise disjoint embedded circles, whose tubular neighborhoods can be weighted independently -- a minimum-marked-length analogue of Colin de Verdiere's classical construction of metrics with prescribed first Laplace eigenvalues. On a surface this mechanism breaks down: finitely many homotopy classes generally cannot be represented by pairwise disjoint simple closed curves, and realizable length data must satisfy global constraints, such as the classical strict-convexity property of the stable norm on the 2-torus (the stable norm is the norm on first homology obtained by minimizing length over each homology class). The authors state the gap explicitly: "We do not know if (4) implies (3) for all 2-dimensional manifolds." A companion paper by the same authors subsequently showed, by a different approximation-based method using isospectral Cayley graphs, that the covering spectrum is not a spectral invariant even in dimension 2 -- there are Sunada-isospectral surfaces of genus 7n+17n+1 with distinct covering spectra -- but that result exhibits Gassmann-Sunada triples that fail the jump condition and leaves the reverse implication untouched. On the geometric ingredient, partial progress exists only for the 2-torus, where the realizable minimum-marked-length data has been characterized through strictly convex stable norms. Whether equality of covering spectra for all invariant metrics forces jump equivalence on surfaces is the open question addressed here.

Problem Statement

Let MM be a closed connected 2-dimensional manifold (a surface) equipped with a free action of a finite group GG by diffeomorphisms, and let H,H′≤GH,H'\le G be subgroups. Consider the two conditions from Theorem 2.2 of de Smit-Gornet-Sutton:

(4) for every GG-invariant Riemannian metric on MM, the quotient surfaces H\MH\backslash M and H′\MH'\backslash M have the same covering spectrum (the covering spectrum of a compact Riemannian surface XX is half the jump set of the filtration of π1(X)\pi_1(X) by the minimum marked length map, i.e. the set of δ>0\delta>0 at which the subgroup generated by the classes of shortest marked length below δ\delta changes, as defined in the background);

(3) (G,H,H′)(G,H,H') is a jump triple, i.e. for all subsets S,T⊆GS,T\subseteq G that are stable under conjugation, ⟨H∩S⟩=⟨H∩T⟩  ⟺  ⟨H′∩S⟩=⟨H′∩T⟩.\langle H\cap S\rangle=\langle H\cap T\rangle \iff \langle H'\cap S\rangle=\langle H'\cap T\rangle.

Question: does (4) imply (3) for all 2-dimensional manifolds MM? The question concerns all closed surfaces, all free actions of a finite group, and all GG-invariant metrics. It holds precisely when equality of covering spectra for all GG-invariant metrics forces jump equivalence for every closed surface and every free action of a finite group, and it fails precisely when there is a non-jump triple (G,H,H′)(G,H,H'), with GG acting freely on a closed surface MM, whose quotient surfaces H\MH\backslash M and H′\MH'\backslash M have equal covering spectra for every GG-invariant metric.

References

  1. Bart de Smit, Ruth Gornet, and Craig J. Sutton, Sunada's method and the covering spectrum, Journal of Differential Geometry 86 (2010), no. 3, 501-537. DOI 10.4310/jdg/1303219428; arXiv:0905.0380; https://arxiv.org/abs/0905.0380
  2. Bart de Smit, Ruth Gornet, and Craig J. Sutton, Isospectral surfaces with distinct covering spectra via Cayley graphs, Geometriae Dedicata 158 (2012), no. 1, 343-352. DOI 10.1007/s10711-011-9637-5; arXiv:1006.5414; https://arxiv.org/abs/1006.5414
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