Published paper

First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

Anatole Katok, Ralf J. Spatzier

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Publication metadata

Journal citation: Publications Mathématiques de l’IHÉS 79, 131–156 (1994)
Publication date: 1994
DOI: 10.1007/BF02698888 ↗

Abstract

Corollary 3.2 is the exponential-mixing input cited in the multiple-mixing branch, but its printed scope is restricted to irreducible cocompact lattices and excludes SO(n,1) and SU(n,1) factors.

Role in dependence graphs

Proof-critical source

Submanifold-genericity of $\mathbb{R}^d$-actions and uniform multiplicative Diophantine approximation

This paper is included only for the following marked statement:

Theorem 3.1 and Corollary 3.2 · printed pp. 140–142 / VOR PDF pp. 10–12Passes from K-finite matrix-coefficient estimates to Sobolev decay for smooth vectors in its restricted cocompact setting.

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Publications Mathématiques de l’IHÉS version of record · volume 79 (1994), pages 131–156

Anatole Katok and Ralf J. Spatzier. First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Publications Mathématiques de l’IHÉS 79 (1994), 131–156.

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Generated August 23, 2026

01Statements2 reported findingsCorrect

The smooth and Hölder first-cohomology rigidity theorems, Livšic theorem, and rigidity applications are correct. The authors later withdrew Proposition 4.9 and supplied a different argument for the affected Hölder theorem; that verified repair preserves the central conclusions and their stated scope.

Theorems 2.9–2.10Correct · verified repair for Theorem 2.9(b)

First-cohomology and higher-rank Livšic conclusions

Printed pages 138–139 and 142–154 · Theorems 2.9–2.10

Part (a) constructs smooth transfer functions by Fourier analysis in the algebraic cases and by exponentially decaying matrix coefficients in the semisimple cases; the Livšic theorem turns vanishing closed-orbit obstructions into Hölder, $C^1$ , or smooth coboundaries. Part (b) is not proved by the printed smoothing proposition, but the authors' erratum proves exponential decay for Hölder matrix coefficients in every standard case, first obtains a distributional solution, then shows that its translates by stable and unstable horospherical groups differ by continuous functions. Since those groups together with $A$ and $M$ generate $G$ , this yields a continuous transfer function, and Theorem 2.10 upgrades it to Hölder. The repair covers exactly the printed conclusion.

Authors’ erratum ↗

Theorems 2.12–2.14Correct

Time-change, local Hölder rigidity, and invariant-volume applications

Printed pages 139 and 154–155 · Theorems 2.12–2.14

A time change is encoded by an $\mathbb R^k$ -valued cocycle; Theorem 2.9 reduces it to a linear homomorphism plus a transfer map, and compactness plus trivial isotropy forces that homomorphism to be an automorphism. Normal hyperbolicity supplies the orbit equivalence for local rigidity. For invariant volume, the logarithmic Jacobian is a cocycle; the repaired Hölder cohomology theorem and the regularity clause of Theorem 2.10 produce a smooth density, while total-volume preservation forces the remaining constant Jacobian to be one. These deductions remain valid after the repair of Theorem 2.9(b).

Version of record and authors’ erratum ↗

02Proofs2 reported findingsContains incorrect or incomplete proofs

The proof is incorrect as printed because Proposition 4.9 is false and is the sole bridge used to deduce Hölder cohomology rigidity from the smooth theorem. The authors' erratum supplies a verified replacement argument. The focal matrix-coefficient estimate is correct within its explicitly narrower cocompact irreducible setting.

Proposition 4.9 and proof of Theorem 2.9(b)Incorrect as written · verified repair

The smooth-approximation proposition is false

Printed pages 152–154 · Proposition 4.9 and proof of Theorem 2.9(b)

The proposition asserts that every continuous cocycle over an arbitrary locally free $\mathbb R^k$ -action can be uniformly approximated by smooth cocycles. The authors explicitly state in their erratum that Proposition 4.9 is wrong and that this invalidates the printed proof of Theorem 2.9(b). The attempted flow-box patching smooths local leafwise potentials while preserving closedness, but it does not establish a globally smooth field with the claimed compatibility. Repair classification: Verified repair. The erratum bypasses smoothing altogether by using exponential decay of Hölder matrix coefficients to construct and regularize the transfer distribution directly.

Authors’ erratum, first paragraph and replacement proof ↗

Theorem 3.1 and Corollary 3.2Correct in scope

The focal Sobolev matrix-coefficient estimate is valid only in its printed scope

Printed pages 140–142 · Theorem 3.1 and Corollary 3.2

For an irreducible unitary representation with discrete kernel, the proof decomposes smooth vectors into $K$ -types, applies Howe's $K$ -finite strongly- $L^p$ bound, and uses Warner's Casimir-eigenvalue and $K$ -type dimension summability to obtain exponential decay controlled by Sobolev norms. Corollary 3.2 then uses irreducibility and cocompactness of $\Gamma$ , Moore ergodicity, and Cowling's representation-uniform exponent when no factor is locally isomorphic to $\mathrm{SO}(n,1)$ or $\mathrm{SU}(n,1)$ . This derivation does not support the broader strong-spectral-gap setting later desired by BEG; that is a downstream source-scope mismatch, not a false statement in this paper. The exact proof chain retains statement-restricted edges to Howe Corollary 7.2, Cowling's integrability theorem, Warner's book, and Moore ergodicity.