arXiv:2603.28535v2

Thermodynamic Formalism for Quasimorphisms: Lattices in Higher Rank Semisimple Lie Groups

Pablo D. Carrasco, Federico Rodriguez-Hertz

math.DSmath.GT37D3522E4037A17

Paper metadata

Original submission: March 30, 2026
Latest revision: September 9, 2026 · version 2
Claimed MSC: 37D35, 22E40, 37A17
arXiv categories: math.DS, math.GT
Comments: Improved presentation and more generality in the main result. Some errors fixed

Abstract

We give a new proof, based on thermodynamic formalism, of a foundational result of Burger and Monod in bounded cohomology. Let $G$ be a noncompact connected semisimple real Lie group with finite center and no factors of real rank one, and let $Γ<G$ be a uniform lattice. We prove that, for every orthogonal representation $π:Γ\to\operatorname{O}_N$ , every $π$ -quasimorphism $L:Γ\to\mathbb{R}^N$ is bounded.

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Audited against arXiv v2

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated October 24, 2026

01Statements4 reported findingsSuspected incorrect statements

The lattice boundedness theorem is correct independently of the proposed proof. The new thermodynamic development contains a false periodic-family count and a false unconditional centralizer-invariance assertion. Its Gibbs construction is not established as printed.

Theorem ANo issue found

The finite-dimensional lattice conclusion has an independent verified justification

PDF pp. 2–3, 5–8, 20–29 · arXiv:2603.28535v2

The reductions through compact and finite kernels are valid: averaging over the finite normal kernel bounds the component perpendicular to its fixed subspace; the projected quasimorphism descends with bounded defect. A finite-index lift removes central torsion, and the irreducible-block decomposition leaves blocks of total real rank at least two. For an almost-simple higher-rank block, comparison injectivity follows from Burger–Monod, Theorem 21, and finite-dimensional affine isometric actions have fixed points by Bader–Furman–Gelander–Monod, Theorem B. For an irreducible block with several simple factors, Burger–Monod's Theorem 16 applies. Any finite-dimensional orthogonal subrepresentation extending through a simple factor is trivial: a connected semisimple group without compact factors has no nontrivial continuous homomorphism to a compact orthogonal group. Consequently their factor modules are precisely the invariant vectors. Their Corollary 24 gives comparison injectivity there; orthogonal projection onto invariant vectors is a coefficient retraction, so comparison is injective for the whole module. Thus $L$ differs boundedly from a genuine cocycle. Its non-invariant component has no almost invariant vectors, by compactness of the finite-dimensional unit sphere. Bader–Furman–Gelander–Monod, Theorem 8.3 and Remark 1.9, decompose this cocycle into finite-dimensional continuous factor cocycles. Their linear parts must be trivial by the same compact-image argument, so the corresponding subspaces inside the non-invariant component vanish. The invariant component is a homomorphism, which vanishes by their Theorem 8.4 and semisimplicity. Hence the cocycle and $L$ are bounded. Finally bounded restrictions on the finitely many commuting blocks bound $L$ on their product using its defect inequality, and finitely many coset representatives recover the original lattice. This independent argument includes irreducible blocks made of rank-one simple factors; it does not incorrectly assume property $(T)$ for those blocks.

Proposition 3.22 and Corollary 3.24Suspected incorrect statement

Orbit representatives do not satisfy the claimed periodic-point growth bound

PDF pp. 16–17, Definition 3.20 and the lower bounds · arXiv:2603.28535v2

Definition 3.20 forbids $T^kx$ from lying in the center plaque of another selected point for every $0\le k<n$ . Take the hyperbolic toral automorphism $T$ with matrix $\left(\begin{smallmatrix}4&1\\3&1\end{smallmatrix}\right)$ . Its center bundle is zero, so it is a center isometry as defined; its irrational unstable lines are dense. Quasiperiodic points are exactly points fixed by $T^n$ , and a maximal separated family contains exactly one point per periodic orbit among them. If $\lambda=(5+\sqrt{21})/2$ , then $h_{\mathrm{top}}(T)=\log\lambda$ and $F_n=\#\operatorname{Fix}(T^n)=\lambda^n+\lambda^{-n}-2$ . For a prime $p$ , the maximal family therefore has cardinality $N_p=(F_p-F_1)/p+F_1$ , so $N_p/e^{ph}\to0$ . No positive constant $C_\delta$ can satisfy the printed lower bound for all $n$ . Taking the zero quasicocycle gives $P=h$ and $Z_n(\mathcal E)=N_n$ , refuting Corollary 3.24's lower bound as well. Possible repair only: change the families to retain separated phases, or incorporate periods and time averaging into a different counting argument. A factor of $n$ cannot be inserted harmlessly in the printed Gibbs proof, which needs uniform constants.

Centralizer-invariance assertion at the end of Section 3Suspected incorrect statement

Commuting with the map does not imply preserving a potential's Gibbs measure

PDF pp. 19–20, paragraph after Corollary 3.31 and its displayed proof · arXiv:2603.28535v2

The assertion says that every commuting homeomorphism $R$ acting isometrically on center leaves preserves the constructed equilibrium measure, without assuming any invariance of the quasicocycle. Use the same toral map as above, let $R=-\operatorname{Id}$ , and take $\varphi(x,y)=\sin(2\pi x)$ with additive quasicocycle $S_n=\sum_{j<n}\varphi\circ T^j$ . This smooth potential has the Bowen property: hyperbolicity bounds the sum of distances of two points in a forward Bowen ball by a uniform geometric-series bound. The two fixed points $q=(1/3,0)$ and $Rq=(2/3,0)$ have potential values $\sqrt3/2$ and $-\sqrt3/2$ . The involution commutes with $T$ and is an isometry for a flat adapted metric; its center-isometry requirement is vacuous. It maps their Bowen balls onto each other. If the Gibbs measure were $R$ -invariant, the two-sided Gibbs estimates would force $\exp(n(\varphi(q)-\varphi(Rq)))=\exp(n\sqrt3)$ to be bounded uniformly in $n$ , a contradiction. In the printed proof, pushing forward changes the weights from $e^{S_n(x)}$ to weights attached to $Rx$ ; these are not generally $e^{S_n(Rx)}$ . Verified restricted repair: require preservation of the potential, or establish that the two potentials are uniformly cohomologous together with the needed uniqueness statement. This is a substantive extra hypothesis, not a notation correction.

Corollaries 3.27–3.31, general Bowen-quasicocycle constructionNot independently verified

The stated general Gibbs construction is not verified by the supplied argument

PDF pp. 17–19 · arXiv:2603.28535v2

The claimed invariant full-support ergodic measure with uniform two-sided Gibbs estimates depends on Corollary 3.24 and Lemma 3.25. The former is false even for a zero potential, as proved above. In the latter, maximality supplies a point near $T^{k(y)}q(y)$ , but the proof then declares that one may assume $k(y)=0$ for an arbitrary already fixed family. Changing the representative's phase changes the measure being estimated; there may be $n+m+2M$ possible phases. The unresolved obligation is a valid uniform weighted counting and time-averaging construction that proves both Gibbs bounds for one invariant measure in the entire stated center-isometry setting. Replacing orbit representatives by phase-separated families is a plausible repair only; no complete replacement proof is supplied here. The same gap affects the asserted canonical-measure uniqueness and the new thermodynamic proof of Theorem A, not the independently verified truth of Theorem A.

02Proofs5 reported findingsSuspected incomplete or incorrect proofs

The algebraic reductions and the final conditional Cartan/Weyl argument are supported. The passage through periodic families and centralizer invariance is not a valid proof of the main theorem as printed.

Proposition 3.22, Corollary 3.24 and Lemma 3.25Suspected incorrect as written

The Gibbs construction loses orbit phases and uses a false uniform lower count

PDF pp. 16–18 · arXiv:2603.28535v2

The toral counterexample in Part 1 satisfies all stated dynamical assumptions and gives only approximately $e^{nh}/n$ orbit representatives, contrary to the displayed lower estimates. The proof of Proposition 3.22 asserts that shadowing a point-separated family produces a family separated in the stronger orbit-representative sense, which need not hold. Lemma 3.25 subsequently replaces independently chosen representatives by suitable phases without controlling how this changes the family measure. These are precisely the inputs used in Corollary 3.27 to construct the invariant Gibbs measure. Plausible repair only: redesign the phase selection and weighted averaging and prove the uniform Gibbs estimates anew. The existing inequalities do not provide that repair.

End of Section 3 and centralizer step in Section 4Suspected incorrect as written

The weighted measure is not functorial under an arbitrary commuting map

PDF pp. 19–20 and p. 25, paragraph after Lemma 4.6 · arXiv:2603.28535v2

The equality between the pushforward of the weighted measures and the measures formed from the image family omits the transformation of $S_n$ . The explicit smooth toral potential in Part 1 disproves the resulting invariance claim. In the lattice application, the general quasicocycle identity does yield a useful additional fact: for fixed $b$ commuting with $a$ , $S(x,ba^n)$ and $S(x,a^nb)$ give $S(xb,a^n)-S(x,a^n)=S(xa^n,b)-S(x,b)+O(1)$ , uniformly bounded in $x,n$ , because compactness of the base gives bounded $S(\cdot,b)$ . Thus the special potentials are cohomologous under that translation. This is a verified algebraic step, but not a complete repair: the required Gibbs existence and uniqueness have the preceding defects, and the noncompact root subgroup in the centralizer need not act isometrically on the center leaves. Its adjoint action expands some zero- $a$ -weight root directions, whereas Section 3's hypothesis requires an isometry. A valid replacement must justify invariance under these particular smooth centralizer translations before applying measure rigidity. The later Ratner-type argument is conditional on that invariance and cannot establish it retroactively.

Lemma 4.6Not independently verified

The twisted Bowen estimate requires control not supplied by nearby fundamental domains

PDF p. 25 · arXiv:2603.28535v2

The assertion is a uniform Bowen bound for every scalar component of the induced twisted quasicocycle. Its proof invokes finite changes of fundamental domains and the twisted defect inequality. But left multiplication by a fixed lattice element changes $L(\gamma)$ to $\pi(\eta)L(\gamma)+O(1)$ , not to $L(\gamma)+O(1)$ . The potentially unbounded term $(\pi(\eta)-\operatorname{Id})L(\gamma)$ remains. Likewise two nearby fiber points with coordinates $A$ and $Ac$ have induced values differing by $(c^{-1}-\operatorname{Id})A^{-1}L([xa^n])$ , although their fiber displacement remains small under all these iterates. The missing obligation is uniform control of these terms before boundedness of $L$ is known. No independent dynamical repair is supplied. The external proof of Theorem A establishes boundedness and hence the Bowen property after the fact, but using it here would replace, rather than validate, the claimed direct proof.

Algebraic reductions and conditional final argumentNo issue found in the reviewed proof

The finite-kernel, product and Weyl reductions are valid

PDF pp. 5–10 and 27–29 · arXiv:2603.28535v2

The finite-kernel projection commutes with the representation and bounds its discarded component; finite-index boundedness passes through finitely many coset representatives. The induced map has uniformly bounded defect by orthogonality. Conditional on the missing Haar/Gibbs step, equality of the means and uniform Gibbs bounds imply uniformly bounded differences on integer wall iterates. Compact fundamental-domain closure bounds the intervening real times. The finitely many wall vectors span the split Cartan, giving a uniform bound on the entire Cartan using only finitely many defect errors. The compact factors in the Cartan decomposition give a uniform error as well. Varying the fiber bounds the non-invariant coefficient component. On the invariant component, homogeneity and Weyl conjugation make the additive Cartan datum Weyl-invariant; each simple reflection kills its coroot direction, and those directions span. Hence the final vanishing follows, once its missing thermodynamic input is supplied.

Proposition 2.4 and the root decompositionSuspected typo

The calculation fixes two harmless inverse and index errors

PDF p. 10, final line of Proposition 2.4; p. 22, definition of the center root space · arXiv:2603.28535v2

The explicit calculation immediately preceding Proposition 2.4 has coefficient $B_1$ , so its stated $\rho((g,B))=B^{-1}$ should be $\rho((g,B))=B$ . The inverse map is not a representation for a nonabelian compact coefficient group; the corrected projection is. All subsequent restrictions to $G\times\{e\}$ have trivial coefficient action either way. In the root decomposition, the center summation condition $\beta(H)>0$ should be $\beta(H)=0$ , uniquely fixed by the adjacent stable and unstable definitions and by $DT_a=e^{-\beta(H)}$ in the next proof. These literal changes do not alter the intended restricted cocycle or the center-isometry argument.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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Paper: arXiv:2603.28535v2
Authors listed: Pablo D. Carrasco, Federico Rodriguez-Hertz
Audit date: October 24, 2026

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Thermodynamic Formalism for Quasimorphisms: Lattices in Higher Rank Semisimple Lie Groups

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The finite-dimensional lattice conclusion has an independent verified justification

Orbit representatives do not satisfy the claimed periodic-point growth bound

Commuting with the map does not imply preserving a potential's Gibbs measure

The stated general Gibbs construction is not verified by the supplied argument

The Gibbs construction loses orbit phases and uses a false uniform lower count

The weighted measure is not functorial under an arbitrary commuting map

The twisted Bowen estimate requires control not supplied by nearby fundamental domains

The finite-kernel, product and Weyl reductions are valid

The calculation fixes two harmless inverse and index errors

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