Non-uniform higher-rank lattices are character rigid

Alon Dogon; Arie Levit; Liam Hanany; Michael Glasner; Yuval Gorfine

REVIEW 2 major objections 6 minor 46 references

Non-uniform higher-rank lattices are character rigid

T0 review · 2 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Every irreducible non-uniform higher-rank lattice is character rigid.

desk verdict A major step: character rigidity for all non-uniform higher-rank lattices in char ≠2, with an honest dependence on a deep external theorem. read the letter →

arxiv 2507.21862 v1 pith:BWOT4UUW submitted 2025-07-29 math.GR math.DSmath.OAmath.RT

Alon Dogon , Michael Glasner , Yuval Gorfine , Liam Hanany , Arie Levit This is my paper · ORCID

classification math.GRmath.DSmath.OAmath.RT MSC 22E4020C0722D1037A15

keywords characterrigidityirreduciblelatticesnon-uniformsemisimplegroupsStuck–Zimmerconjectureinvariantrandomsubgroupscongruencecharactersmixingtraces

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that every irreducible non-uniform lattice in a semisimple group of rank at least two is character rigid, provided the underlying local fields do not have characteristic 2. Character rigidity means any indecomposable trace — a normalized, conjugation-invariant positive-definite function — is either built from a finite-dimensional unitary representation or vanishes outside the center. The result closes the non-uniform case of the Stuck–Zimmer conjecture: every ergodic probability-measure-preserving action of such a lattice is either essentially transitive or has all non-central elements acting with fixed-point sets of measure zero. It also forces every ergodic invariant random subgroup to be either supported on the conjugacy class of a finite-index subgroup or concentrated on a central subgroup. The proof works by establishing a mixing phenomenon: any weakly mixing trace decays along elements of a split torus, uniformly over unipotent directions.

What carries the argument

The load-bearing mechanism is a dichotomy for characters of the solvable arithmetic semidirect product $M = \langle a \rangle \ltimes U(R)$, where $U$ is the unipotent radical of a minimal parabolic and $a$ generates a Zariski-dense subgroup of a split torus. Proposition 5.6 says each character of $M$ is either induced from $U$ (hence infinite-dimensional and vanishing off $U$) or restricts to a congruence trace on $U$ (hence finite-dimensional). Congruence characters are detected dynamically: a character of $U(R)$ is congruence exactly when its orbit under the dual action of $\langle a \rangle$ is finite (Proposition 5.5). Theorem 6.1 then uses the theorem that opposite congruence subgroups $U(I)$ and $V(I)$ together generate a finite-index subgroup of $\Gamma$; the contradiction argument shows that a non-mixing component on the split torus would produce a common invariant vector for these two subgroups and hence a finite-dimensional subrepresentation, contradicting weak mixing.

What would settle it

Compute the subgroup generated by the two opposite congruence subgroups U(I) and V(I) inside G(R) for a rank-one isotropic group over a global field of characteristic 2: if some non-zero ideal I gives a proper subgroup of infinite index, the generation theorem fails and the main mixing argument breaks. Equivalently, produce a weakly mixing trace φ on G(R) with limsup of |φ(a^i u)| not tending to zero for a Zariski-dense a; this would refute Theorem 6.1 and with it the claimed character rigidity.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is a decay theorem (Theorem 6.1) for traces of the arithmetic lattices $\Gamma = G(R)$ that arise from a global field $F$ of characteristic $\neq 2$, a rank-one isotropic algebraic group $G$, and a ring of $S$-integers $R$ with $|S| \geq 2$. The theorem says that every weakly mixing trace $\varphi$ of $\Gamma$ satisfies $\lim_{i \to \infty} \sup_{u \in U(R)} |\varphi(a^i u)| = 0$ for an element $a$ generating a Zariski-dense subgroup of a split torus and for $U$ the unipotent radical of a minimal parabolic subgroup. From this, character rigidity of $\Gamma$ is derived by two routes: a self-contained Bruhat-decomposition argument using a generalized Bekka vanishing lemma, and a softer argument through charmenability that reduces the problem to the known rigidity of $SL_2(R)$. Combined with the arithmeticity theorem and existing property-(T) results, this yields Theorem 1.3: all non-uniform higher-rank irreducible lattices in characteristic $\neq 2$, and all such lattices with a property-(T) factor, are character rigid.

Load-bearing premise

The argument rests on a known generation theorem: the congruence subgroups attached to two opposite unipotent subgroups together generate a finite-index subgroup of the lattice; should that theorem fail, for instance in characteristic 2, the key mixing contradiction would collapse.

Editorial extensions

If this is right

Every ergodic probability-measure-preserving action of such a lattice is either essentially transitive or satisfies $\mu(\mathrm{Fix}(g)) = 0$ for every non-central $g$ (Corollary 1.4).
Every ergodic invariant random subgroup of such a lattice is either uniform on the conjugacy class of a finite-index subgroup or Dirac on a central subgroup.
The lattices considered are charfinite, so their uniformly recurrent subgroups and unitary representations inherit the corresponding rigidity properties.
The same character rigidity and stabilizer rigidity conclusions hold for irreducible lattices in higher-rank semisimple Lie groups with arbitrary, possibly infinite, center (Theorem 1.5).
The non-uniform case of the Stuck–Zimmer conjecture is now settled for all characteristics other than 2.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The characteristic-2 restriction appears to be an artifact of the opposite-congruence generation theorem; if that theorem holds over characteristic-2 fields, the same proof should remove the restriction entirely.
The uniform decay in Theorem 6.1 may admit explicit rates tied to Dirichlet units and the filtration by congruence subgroups, giving quantitative versions of character rigidity not stated in the paper.
The congruence/non-congruence dichotomy for characters of unipotent arithmetic groups suggests a similar dichotomy for invariant random subgroups, linking character rigidity to the congruence subgroup property.
A natural test case beyond the paper's scope would be SL_2 over rings of S-integers in characteristic 2, where the same character classification should either hold or produce a counterexample that pinpoints the failure of the generation theorem.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on 2507.21862. The paper is the real thing: it proves character rigidity for every non-uniform irreducible higher-rank lattice in semisimple groups of characteristic different from 2, which resolves the Stuck–Zimmer conjecture for that entire class. The main new engine is Theorem 6.1, a mixing theorem for weakly mixing traces on the F-split torus of an arithmetic rank-one group, and the paper gives two independent proofs of the main theorem, both running through that result. The congruence-character machinery in Section 5 is a genuine technical contribution, not a repackaging.

The authors are also honest about the boundary: the only place char(F)≠2 enters is Raghunathan–Venkataramana's theorem (Thm 6.2), and they say explicitly they don't know whether it holds in characteristic 2. That's exactly the right way to handle a restriction. The reduction in Proposition 3.5, showing that every lattice in the statement is commensurable (up to central extension) to a group G(R), is careful and makes the arithmetic setup credible.

Now the soft spots, in proportion. The central theorem is exactly as strong as the quoted theorem of Raghunathan and Venkataramana. If that theorem fails, both proofs collapse and there is no fallback. That's a dependence, not a defect, but it deserves to be explicit. More concretely, Theorem 6.1 as stated invokes Theorem 6.2, whose hypothesis is sum of local ranks at least 2. The Standing Assumptions only give rank_F(G)=1 and |S|≥2, which is not enough in general (a quaternion group over Q split at exactly one place in S is a counterexample to the Standing Assumptions implying the sum). In the cases that matter for Theorem 1.2, the condition follows from rank(H)≥2 via Proposition 3.5, so it's fixable by adding a clause to the Standing Assumptions or to Theorem 6.1. A referee should chase that. Minor: Lemmas 2.6 and 8.3 are left to the reader (both are routine), and the proof of Lemma 6.7 claims SOT convergence where the argument really needs trace convergence; the trace statement is true and sufficient, so this is a small presentation slip.

Who should read it: anyone working on characters, invariant random subgroups, or stabilizer rigidity for lattices. It deserves a serious referee. Send it out, but the referee should ask the authors to make the local-rank-sum hypothesis explicit wherever Theorem 6.2 is used. I'd take the result to a reading group and would cite it as the resolution of the non-uniform case.

Referee Report

2 major / 6 minor

Summary. The paper proves character rigidity for all irreducible non-uniform lattices in semisimple groups of rank at least two, under the assumptions that the ambient local fields have characteristic different from 2 and, in the case of Theorem 1.2, that the semisimple group admits an almost simple rank-one factor. The proof reduces to arithmetic lattices Γ = G(R) via Margulis' arithmeticity theorem, then proves a new mixing theorem (Theorem 6.1): every weakly mixing trace on Γ restricts to a mixing trace on an F-split torus, with uniform decay along elements of the form a^i u. The key ingredients are a detailed character theory of solvable arithmetic groups (§5, congruence characters and induced characters), the Raghunathan–Venkataramana finite-index generation theorem (Theorem 6.2), and a careful Bruhat-decomposition argument in rank one. Two independent proofs of Theorem 1.2 are given: one self-contained using the generalized Bekka lemma, and one using charmenability and reduction to the Peterson–Thom theorem for SL2. The paper also derives stabilizer rigidity for probability-measure-preserving actions (Corollary 1.4), invariant random subgroup rigidity, and an extension to lattices with infinite center (Theorem 1.5).

Significance. If the main theorem is correct, this is a substantial advance on the Stuck–Zimmer conjecture: it removes, for all non-uniform higher-rank lattices in characteristic different from 2, the previous dependence on a Kazhdan property-(T) factor. The paper includes two independent proofs of the central result, which strengthens confidence in the argument, and it treats the non-property-(T) case (products of rank-one factors) that was the main open territory. The dependence on the external Raghunathan–Venkataramana theorem is stated honestly, including the explicit caveat that characteristic 2 is excluded precisely because that theorem is not known there. The proof is parameter-free and has no fitted constants or normalization tricks; the main line is structurally coherent and the Berkovich-style reductions through commensurability and central extensions are handled abstractly. The derived applications to stabilizers and invariant random subgroups are natural and clearly explained.

major comments (2)

[§6, proof of Theorem 6.1] The proof of Theorem 6.1 invokes Theorem 6.2 at the final step, where the subgroups U_n and V_n generate a finite-index subgroup of Γ = G(R). Theorem 6.2 requires, according to Remark 6.3, that ∑_{v∈S} rank_{F_v}(G) ≥ 2. This condition is not part of the Standing Assumptions of §3.1, which only require rank_F(G) = 1, |S| ≥ 2, and char(F) ≠ 2. In the applications obtained through Proposition 3.5 the condition does follow from rank(H) ≥ 2, because at least two local places contribute noncompact factors; however, as Theorem 6.1 is stated, it covers cases where the hypothesis of Theorem 6.2 is not known to hold. The mismatch is load-bearing, since the finite-index conclusion is precisely what produces the finite-dimensional subrepresentation contradicting weak mixing. The authors should either add the local-rank-sum condition to the Standing Assumptions/Theorem 6.1 or explicitly verify it before the invocation of Theorem 6.2.
[§8.3, Lemma 8.4] The proof of Lemma 8.4 contains an invalid inference: it asserts that because Γ is finitely generated and separated by its finite-dimensional unitary representations, it is residually finite. This does not follow, since a finite-dimensional unitary representation of a finitely generated group need not have finite image. Separation by such representations therefore does not yield finite quotients separating points. Lemma 8.4 is used in Lemma 8.5 to establish that the lattice Λ in Theorem 1.5 has at most countably many finite-dimensional unitary representations, and hence Theorem 1.5 depends on it. This does not affect the main Theorem 1.2, whose proof uses [4, Proposition 7.1] directly, but it is a genuine gap in the statement of Theorem 1.5 and should be repaired with a valid argument or by restricting the scope of the lemma.

minor comments (6)

[§6, Lemma 6.7] The proof says that the sequence (p∧q)∨p_n − p_n converges to 0 in the strong operator topology, but what is actually established is trace convergence. Since the subsequent argument only uses trace convergence and the normality of τ, the statement and proof should be adjusted accordingly.
[§2.3, Lemma 2.6] The proof is compressed: the verification that H_mix is a closed invariant subspace and the uniqueness of the decomposition are left to the reader. These facts are straightforward, but given that the lemma is used later for the restriction to a torus, a few lines of detail would improve readability.
[§8.3, Lemma 8.3] The proof is omitted with a reference to a 'straightforward verification' by induction and restriction. Since the lemma is used in Proposition 4.2, a short proof or an explicit reference would be preferable.
[§8.1, proof of Theorem 1.3] In the case where H has no factor with Kazhdan's property (T), the proof silently reduces to Theorem 1.2, which requires an almost simple rank-one factor. The authors should state the standard fact they are using: a simple factor over a local field without property (T) has rank one, so the hypothesis of Theorem 1.2 is indeed satisfied.
[Throughout] There are several typographical errors, e.g. 'Apriori' in §3, 'assocated' and 'combitation' in Proposition 5.8, and 'Induction' for 'inductions' in Lemma 8.3. These should be corrected in the final version.
[§2.2, proof of Proposition 2.4] The notation ψ_i(·) = φ_i(π_{φ_i}(·)p_i) is ambiguous, since φ_i is first introduced as a function on the group. Clarifying the identification of the trace on the von Neumann algebra with the function on the group would help the reader.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof reduces to independent external theorems and the self-citations are technical and non-load-bearing.

full rationale

I walked the derivation chain of Theorem 1.2. The theorem is reduced via Margulis arithmeticity (Proposition 3.5) to the arithmetic group Gamma = G(R), then character rigidity for Gamma is proved in two ways, both relying on the mixing theorem (Theorem 6.1). The proof of Theorem 6.1 uses the Raghunathan-Venkataramana theorem (Theorem 6.2), which is an external deep result about finite-index generation by opposite unipotent congruence subgroups, with the characteristic-2 restriction explicitly flagged as the only place it is used. The first proof of Theorem 1.2 combines Theorem 6.1 with the generalized Bekka lemma quoted from [14], a general vanishing lemma about traces that does not assume character rigidity. The second proof uses charmenability from [3] and the SL2(R) character rigidity of Peterson-Thom [34], both external. Self-citations appear, notably [5] for an ICC fact, [14] for the Bekka-type lemma, and [29] for a lemma on characters of semidirect products, but each is a technical standalone result with stated general assumptions not including the target theorem. No fitted parameter is introduced, no prediction is a restatement of an input, and no load-bearing claim is justified only by a self-citation chain. The skeptical concern about the hypotheses of Theorem 6.2 is a genuine correctness/robustness question, not a circularity, since the theorem is cited as an independent external result. Overall, the derivation is self-contained relative to well-documented external inputs, and the characteristic-2 caveat is an honest boundary rather than a disguised conclusion.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The paper relies on a series of established theorems: Thoma's theorem, Choquet theory, Borel-Harish-Chandra, Margulis arithmeticity, Dirichlet's unit theorem, Raghunathan-Venkataramana, Tits classification, and a finiteness result for finite-dimensional representations. It introduces no free parameters and no new postulated entities. The hypotheses of the main theorem (characteristic not 2, at least two places, rank-one factor) are explicit scope conditions, not hidden assumptions.

assumptions (8)

standard math Thoma's theorem: traces on countable groups correspond to tracial von Neumann algebra representations (Theorem 2.2).
Used throughout Section 2 to identify characters with extremal traces and to relate GNS representations to von Neumann algebras.
standard math Choquet theory: the compact convex set of traces has a unique integral decomposition over characters (Bochner transform).
Used in Sections 2 and 5 to decompose restrictions of traces to unipotent subgroups.
standard math Borel-Harish-Chandra theorem: S-arithmetic subgroups are lattices, uniform if and only if the group is F-anisotropic (Theorem 3.3).
Used in Section 3 to justify that Gamma = G(R) is a non-uniform lattice.
standard math Margulis arithmeticity theorem: every irreducible lattice in rank at least 2 is S-arithmetic up to finite central extension (Theorem 3.4).
Used in Proposition 3.5 to reduce the general lattice setting to the arithmetic standing assumptions.
standard math Dirichlet-Hasse-Chevalley unit theorem: the unit group of R = F(S) has rank |S| - 1 (Remark 3.6).
Provides the infinite order unit a in A(R) used throughout Sections 5 to 7; this is where non-uniformity enters.
standard math Raghunathan-Venkataramana theorem: for every non-zero ideal I, the subgroups U(I) and V(I) generate a finite-index subgroup of Gamma (Theorem 6.2).
Used in the proof of Theorem 6.1 to obtain the finite-dimensional subrepresentation contradiction; the characteristic not 2 assumption is needed only here.
standard math Tits classification of semisimple algebraic groups by Tits index and anisotropic kernel (Theorem A.1).
Used in Appendix A to list the lattices in characteristic zero that are not covered by property (T).
standard math Lattices in higher-rank semisimple groups have at most countably many finite-dimensional unitary representations ([4, Proposition 7.1]).
Used in Section 4 to prove that character rigidity is invariant under commensurability and central extensions.

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Cite this review

Pith. "Pith review of Non-uniform higher-rank lattices are character rigid." pith.science (2026). https://pith.science/paper/BWOT4UUW

@misc{pith2026250721862,
  author       = {Pith},
  title        = {Pith review of: Non-uniform higher-rank lattices are character rigid},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BWOT4UUW}},
  note         = {Machine review of arXiv:2507.21862}
}

read the original abstract

We establish character rigidity for all non-uniform higher-rank irreducible lattices in semisimple groups of characteristic other than 2. This implies stabilizer rigidity for probability measure preserving actions and rigidity of invariant random subgroups, confirming a conjecture of Stuck and Zimmer for non-uniform lattices in full generality.

Discussion (0). Continue with ORCID to comment.

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

3, 465–488

Mikl´ os Ab´ ert, Yair Glasner, and B´ alint Vir´ ag,Kesten ’s theorem for invariant random subgroups, Duke Mathematical Journal 163 (2014), no. 3, 465–488

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work page 2023

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Uri Bader, R´ emi Boutonnet, Cyril Houdayer, and Jesse Peterson,Charmenability of arithmetic groups of product type , Invent. Math. 229 (2022), no. 3, 929–985. MR 4462622

work page 2022

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