Is Mostow rigidity an instance of the trace identity or independent of it?

Object. Two ways an archimedean magnitude can be determined by discrete data.

The first is the trace identity, tr(T) = Σ λₙ = ∫ K(x,x): the trace of an operator computed in its spectral presentation and in its geometric one must agree, because the trace is conjugation-invariant. Its instances include Poisson summation, the Selberg and Gutzwiller trace formulas, Grothendieck–Lefschetz, Duistermaat–Heckman, the heat-kernel index theorems, and Weil's explicit formula.

The second is rigidity. Mostow's theorem says a closed hyperbolic manifold of dimension at least three admits exactly one hyperbolic structure, so its volume — a real number with no obvious combinatorial definition — is a topological invariant. That is a determination of a magnitude by discrete data, and on its face it involves no operator, no spectrum, and no sum over anything.

Presenting problem (the friction). Everything else in this family turned out to be one theorem. Rigidity does not obviously fit, and the two possible resolutions have different consequences.

But the independence is not clean either. For a closed hyperbolic 3-manifold, Weyl's law gives the eigenvalue counting function N(λ) ~ vol·λ^{3/2}/(6π²), so the volume is read off the leading asymptotics of the Laplace spectrum — which is the trace identity at small time. The volume is therefore a spectral quantity whether or not Mostow is used to get at it, and Mostow's contribution is that it is also a topological one.

So the question is whether rigidity is a separate mechanism for determining magnitudes, or a theorem about when a quantity the trace identity already computes happens to be a homeomorphism invariant.

History. Mostow's proof and Margulis's later arguments run through quasiconformal maps and ergodicity of the geodesic flow at the sphere at infinity — no trace formula anywhere. The Selberg trace formula for hyperbolic manifolds relates the spectrum to the length spectrum of closed geodesics with the volume appearing in the identity contribution, so the two objects are certainly in contact. Gromov's proof by simplicial volume gives a third route, combinatorial and again trace-free.

Differential. 1. Inside. Rigidity is not a mechanism for determining a magnitude; the spectrum already determines the volume by Weyl, and rigidity is the separate statement that the geometric structure is unique. There is one mechanism, not two. 2. Independent. Rigidity determines the volume from topology alone, with no operator required, and the Weyl-law route is a coincidence of hyperbolic geometry rather than the general reason. Simplicial volume is the evidence: it computes the same number with no analysis at all. 3. A converse rather than a mechanism. The trace identity computes a magnitude from a structure; rigidity says the structure is unique given weaker data. Those are different directions, and grouping them by "determines a magnitude" is grouping by output rather than by operation. 4. Inside, but only in the hyperbolic case. Rigidity phenomena elsewhere — Margulis superrigidity, Mostow in other settings — have no accompanying Weyl law, so the answer depends on the instance and there is no general fact.

Discriminating tests. Compute the volume of a hyperbolic 3-manifold two ways — from the leading spectral asymptotics and from an ideal triangulation — and check that they agree, which settles whether the spectral route is available at all in practice. Then ask whether simplicial volume can be extracted from any spectral asymptotic: if it can, branch 1; if it provably cannot, branch 2 or 3. For branch 4, find a rigidity theorem in a setting with no Laplacian and check whether any magnitude is determined there.

Stakes. Whether the machinery for reaching magnitudes is one theorem with a boundary, or a short list of genuinely different mechanisms. The first is a classification; the second is a catalogue, and only the first tells you where to look when a new magnitude resists.

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Update (2026-09-13): rigidity is not an invariant, which changes what the comparison is between.

The case sets two mechanisms side by side as two ways a magnitude can be determined by discrete data. Independent work found that one of the two is not the same kind of object as the other.

A trace identity is a computation. Rigidity is an assertion about computations. The trace identity produces a number by evaluating one quantity two ways. Rigidity asserts that some invariant is complete — that the map from structures to invariants is injective. It is second-order: about invariants rather than being one.

The dimension count makes it concrete. Hyperbolic structures on a genus-g surface form a deformation space of dimension 6g − 6 — 6, 12, 24, 54 for g = 2, 3, 5, 10 — a continuum of inequivalent structures. In dimension three and above the deformation space is a point. The number is not computed by the rigidity statement; the rigidity statement says there is only one thing it could be, after which any invariant at all determines it.

Which supplies the case's missing asymmetry. The two mechanisms were treated as parallel routes. They are not parallel: rigidity licenses a determination and does not perform one. That is why the volume is computed by an entirely separate machine (a sum of dilogarithms over an ideal triangulation) while rigidity only guarantees the answer is well defined.

And it generalises, with a criterion. In representation-theoretic settings the same property has an exact test: a first-order symmetry-breaking formula is rigid precisely when the breaking operator appears with multiplicity one in R ⊗ R̄, since by Wigner–Eckart that multiplicity is the number of independent reduced matrix elements. Multiplicity one means the space of possible formulas is a point — the same statement as a deformation space of dimension zero, in a setting with no geometry at all.

Effect on the differential. The branch holding that the two are instances of one mechanism is weakened: a second-order statement and a first-order computation cannot be instances of a common mechanism without an argument that was never given. The sharper question is whether every rigidity statement is accompanied by a separate determination machine, and whether that machine is always a trace. If it is, the two are related as licence and execution rather than as rivals, and the case's framing is the thing that needs replacing.

Discriminating test. Take three rigidity statements from unrelated fields — Mostow, the multiplicity-one criterion above, and the uniqueness of the maximal-dimension supergravity — and in each ask: what actually computes the number, and is it a trace? A consistent answer settles the relation; an inconsistent one says rigidity does not have a single companion mechanism.