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REVIEW 3 major objections 5 minor 53 references

Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\"{u}ck-Zimmer Theorem

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Pith's one-line read This paper proves that a non-lattice discrete subgroup Γ of a higher-rank simple Lie group forces the injectivity radius of its locally symmetric space X/Γ to grow at least like c log-log-log-log r on growing balls, and characterizes lattic

desk verdict Real results, but the central tool is a black box from an unpublished companion, and the advertised Γ-independence of c_G in Theorem 2.4(b) is not established by the proof. read the letter →

arxiv 2608.01382 v1 pith:PHXRKUEP submitted 2026-08-02 math.DS

Ilya Gekhtman , Simon Machado , Omri Solan , Yuval Yifrach This is my paper · ORCID
classification math.DS MSC 22E4037A1753C35
keywords injectivityradiushigher-rankLiegroupsalmoststationarymeasuresrandomwalksonhomogeneousspacesStück–Zimmertheoremlatticerigidityproperty(T)locallysymmetric
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 a quantitative rigidity theorem for discrete subgroups of higher-rank simple Lie groups. It shows that a non-lattice subgroup Γ forces the injectivity radius of the associated locally symmetric space to grow at least like $c\log^{(4)} r$ along balls of radius $r$, with a fixed positive proportion of random-walk steps landing at such points. Equivalently, if the maximal injectivity radius on balls grows slower than four iterated logarithms, Γ must be a lattice — a sharp dichotomy with no intermediate growth rate. The proof replaces the classical passage to invariant limit measures with almost stationary Cesàro averages, yielding an explicit gap phenomenon: an almost invariant probability measure on $G/\Gamma$ exists only when Γ is a lattice. As a by-product, the method gives a one-page proof of the Stück–Zimmer theorem.

What carries the argument

The load-bearing mechanism is the 'almost Nevo–Zimmer dichotomy' (Theorem 3.5), imported from the authors' companion manuscript [26]: an $\epsilon$-almost stationary probability measure on a $G$-space is either almost invariant (up to an error polynomially small in $|\log\epsilon|^{-1}$) or has a definite proportion of its mass concentrated on points whose stabilizers are approximately contained in a proper parabolic subgroup. The paper feeds into this dichotomy the Cesàro averages $\nu_n = \frac1n\sum_{i=1}^n \mu^{*i}*\delta_x$, which are automatically $2/n$-almost stationary with no limit passage. Case (a) is ruled out by Lemma 3.6 (Theorem B), which uses property (T) in its $L^1$ form to

What would settle it

Exhibit a discrete non-lattice subgroup $\Gamma$ of $SL(3,\mathbb{R})$ and a sequence $r_n\to\infty$ for which the maximal injectivity radius on the ball of radius $r_n$ grows slower than $c\log^{(4)} r_n$ for every $c>0$; Corollary 1.1 asserts no such $\Gamma$ exists. A computable candidate would be a finitely generated free subgroup whose quotient's injectivity radius can be numerically traced along random-walk orbits, checking whether the iterated-logarithm growth is violated.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2.4: for every connected simple Lie group $G$ with real rank at least 2 and every discrete infinite-covolume subgroup $\Gamma$, there are constants $c>0$ and $r_0$ such that for every $r\ge r_0$ the ball of radius $r$ in $X/\Gamma$ contains a point of injectivity radius at least $c\log^{(4)} r$, and a definite proportion $c_G$ of the Cesàro random-walk steps visit such points. Corollary 1.1 restates this as an exact dichotomy: either $\Gamma$ is a lattice and the injectivity radius is bounded, or the maximal injectivity radius on balls grows at least like a positive multiple of $\log^{(4)} r$. The paper also establishes a lattice criterion (Theorem B): if a proba

Load-bearing premise

The proof rests on the almost Nevo–Zimmer dichotomy for $\epsilon$-almost stationary measures (Theorem 3.5 of this paper), which is proved in the companion manuscript [26]; if that dichotomy fails in the stated form—for instance, if an almost stationary measure could sit in a third case between almost invariance and almost projective factors—the main theorems of this paper do not follow.

Editorial extensions

If this is right

  • Corollary 1.1 is a dichotomy with no intermediate growth: a discrete subgroup $\Gamma$ is a lattice if and only if the maximal injectivity radius on balls of radius $r$ is $o(\log^{(4)} r)$.
  • The lattice criterion (Theorem B) gives a new gap phenomenon: on $G/\Gamma$ for non-lattice $\Gamma$, no probability measure can be almost invariant below a threshold $\epsilon_0(G,\Gamma)$; approximate invariance forces exact invariance.
  • Theorem 2.4 quantitatively strengthens Fraczyk–Gelander's resolution of Margulis' conjecture, providing explicit iterated-logarithmic growth of unbounded injectivity radius.
  • The Stück–Zimmer theorem follows by a one-page argument from the almost stationary dichotomy, the reduction to invariant random subgroups, and the lattice criterion.
  • The paper announces (in work in preparation) that the constant $c_G$ can be taken arbitrarily close to 1, yielding the first explicit threshold in Benjamini–Schramm convergence for arbitrary sequences of lattices.

Reading between the lines

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

  • The four iterated logarithms are almost certainly not optimal; the explicit accounting for each logarithm suggests a roadmap for shaving them off, and a growth of order $\log r$ might be the true threshold.
  • The same almost-stationary machinery should apply to other geometric or spectral quantities previously accessible only through limit measures, such as effective normality of subgroups or quantitative Betti-number convergence without congruence structure.
  • Because the lattice criterion only needs weak almost invariance of a single measure, its contrapositive provides a potential numerical certificate: an explicit measure on $G/\Gamma$ with $W^f_1$-distance below $\epsilon_0$ would prove $\Gamma$ is not a lattice.
  • If the companion manuscript's dichotomy is extended to cover free actions or a wider class of step measures, the Stück–Zimmer theorem and the injectivity-radius bounds would carry over to settings not covered by the current statement.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper develops quantitative versions of higher-rank rigidity results for discrete subgroups of noncompact simple Lie groups. Its main theorem (Theorem 2.4) asserts that if a discrete subgroup Γ has infinite covolume, then in every sufficiently large ball of X/Γ there is a point with injectivity radius at least c log^(4) r, and a positive proportion of the points visited by a random walk of length r have this property. Corollary 1.1 derives a lattice-rigidity statement: if maximal injectivity radius on balls grows slower than log^(4) r, then Γ is a lattice. The paper also proves a new characterization of lattices via the existence of an approximately invariant measure (Theorem B / Lemma 3.6) and gives a one-page proof of the Stück–Zimmer theorem for ergodic actions of higher-rank simple groups with property (T), replacing the traditional Nevo–Zimmer/intermediate-factor machinery with an effective almost-stationary structure theorem. The appendices contain substantial analytic inputs: heat-kernel log-Lipschitz estimates, random-walk concentration, and a convolution-power Margulis-type inequality, all proved in the text.

Significance. If the main theorem holds as stated, it is a substantial quantitative advance over the qualitative result of Fraczyk–Gelander, and the short Stück–Zimmer proof is an elegant demonstration of the power of the almost-stationary method. The paper is careful in many places: the random-walk, heat-kernel, and discreteness-radius appendices are detailed and self-contained, and the geometric part of the argument is written with a clear structural strategy. The main obstruction to acceptance is the heavy reliance on the companion manuscript [26], which is cited as 'Manuscript in preparation' and is not available. A second, genuinely internal issue is that the constant c_G in Theorem 2.4(b) is not proved to depend only on G; the proof gives c_G = c_G(Γ). Both issues are load-bearing for the stated main theorem, but both are in principle repairable. The central geometric ideas appear sound for the weaker formulation with constants depending on Γ.

major comments (3)
  1. [§3, Theorem 3.5; §5, Lemma 5.16] The main engine of the paper is Theorem 3.5, the 'almost Nevo–Zimmer' dichotomy, which is stated as a black box from [26, Theorem 1.10], an unpublished companion manuscript. It is used directly in the proof of Theorem 2.4 to split into the almost-invariant case and the almost-projective-factor case, and it is also needed for the Stück–Zimmer proof through Theorem 2.4. In addition, Lemma 5.16 begins with 'The fact that ν̃_m can be written in this form follows from the almost Furstenberg decomposition [26]', so the almost-Furstenberg decomposition is another imported, unproved input. As submitted, the central theorem cannot be independently checked. I am not questioning the validity of the companion results, but for a published proof the dependency must be verifiable. The authors should either include the relevant statements and proofs of the companion results, post the companion manuscrip
  2. [Theorem 2.4(b) and proof, Eq. (3.20)] Theorem 2.4(b) asserts that the proportion c_G depends only on G. In the proof, however, the constant η is fixed as η=(ε0/(2C))^4, where ε0=ε0(Γ) is the threshold from Lemma 3.6. Lemma 3.6 states ε0=ε0(Γ), and its proof obtains ε0 from property (T_B) applied to L^1(G/Γ,m_{G/Γ}); the resulting Kazhdan-type constant depends on the quasi-invariant measure, hence on Γ. Equation (3.20) then sets c_G := (1/32)(ε0/2)^4/(2C), so c_G inherits this Γ-dependence. Nothing in Sections 3–4 gives a uniform positive lower bound for ε0(Γ) over all discrete infinite-covolume subgroups Γ. Thus the assertion 'where c_G > 0 depends only on G' is not proved. The argument establishes at most a constant c(G,Γ). This is load-bearing for the statement of part (b), but not for the main Theorem A or the Stück–Zimmer application, which only require a positive proportion for each fixed Γ. I recommend either proving t
  3. [§5, Lemma 5.13 and proof of Lemma 5.1] The proof of Lemma 5.1 goes through Lemma 5.13, whose hypothesis includes a norm-δ, P-invariant measure λ. The construction of λθ in Lemma 5.16 gives measures that are m^{-1/2}-P-almost invariant, not exactly P-invariant. In the proof of Lemma 5.1, this is accounted for by taking δ=m^{-1/2}; however, Lemma 5.13 is stated and proved under exact P-invariance, and the perturbation argument that would justify applying it to an almost invariant measure is only sketched. I am not claiming the gap is fatal, but the formal status of the 'almost P-invariance to exact invariance' passage should be stated clearly as an approximation step, with the resulting error terms tracked through Claims 5.10 and 5.11.
minor comments (5)
  1. [Abstract] The abstract's criterion for lattices uses W_1^b(gν,ν) ≤ ε0, whereas Theorem B in §1.3 and Lemma 3.6 require a stronger condition involving the observable f_{ν,g} = dν/d(gν), plus Lipschitz and bounded-density assumptions. The abstract should match the precise statement, or explicitly call itself a simplified version.
  2. [§6, proof of Theorem 1.2] The sentence 'Since δ_{e} is a G-invariant measure and since ν was assumed ergodic, we obtain that ν=δ_{e} as desired' appears twice verbatim. Please remove the duplication.
  3. [Throughout] There are small typos: 'Cézaro' should be 'Cesàro', 'at last on' should be 'for at least one of', and 'injection radius' should be 'injectivity radius' where it appears.
  4. [§2, Definition 2.3 and §3.5] The Lambert W-function is introduced and not used in an essential way in the main text; the explicit formula following Eq. (3.16) is hard to read. It would help to state the bound in asymptotic form directly, which is already done in Eq. (3.17).
  5. [References] Reference [50] appears to cite a Math StackExchange user name ('stephantu') rather than a standard citable source. If the fact is standard, replacing this reference by a textbook or by a proof in the appendix would be more appropriate.

Circularity Check

2 steps flagged · score 6.0 of 10

Main engine Theorem 3.5 is restated verbatim from the authors' unpublished companion [26], and Lemma 5.16's almost Furstenberg decomposition is likewise imported from [26]; the central claim rests on a load-bearing self-citation chain.

  1. self citation load bearing [Section 1.5 (Method); Section 3, Theorem 3.5]
    "The measure ν n is then fed into the main theorem of [26], which in the form we use reads, informally: ... The precise statement, with the definition of an almost projective factor, is Theorem 3.5 in Section 2; see [26, Section 1.2] for a discussion of it."

    Theorem 3.5 is not proved here; the paper says "Now we can re-state [26, Theorem 1.10]". The proof of Theorem 2.4 applies Theorem 3.5 to the Cesàro almost-stationary measure νn and relies on its dichotomy (a)/(b) as the only route to the conclusion. [26] is an unpublished manuscript in preparation by the same four authors. Thus the main quantitative injectivity-radius claim is derived from a theorem that is itself unverified and inaccessible; the derivation chain reduces to this self-citation rather than to an independently checkable statement.

  2. self citation load bearing [Section 5.3, Lemma 5.16 and Corollary 5.18]
    "The fact that ˜νm can be written in this form follows from the almost Furstenberg decomposition [26]."

    Lemma 5.16 is the bridge that turns the pointwise almost-factor hypothesis of Lemma 3.7 into a family of norm m^{-1/2}-P-almost invariant measures λθ via the 'almost Furstenberg decomposition' cited to [26]. Corollary 5.18 and the proof of Lemma 3.7 use exactly this decomposition to produce the positive-measure set satisfying (3.15), which is the heart of case (b) of Theorem 2.4. So the second branch of the main dichotomy again rests on the same unpublished same-author manuscript, making the self-citation chain load-bearing in both halves of the proof.

full rationale

The paper has substantial independent content: Theorem B/Lemma 3.6 is proved in Section 4 using property (T) in L^1 (Bader–Furman–Gelander–Monod), the group-theoretic estimates of Section 5 and the heat-kernel/GLM appendices are self-contained, and the injectivity-radius statement is not literally an input of [26]. For that reason this is not an 8 or 10 circularity. However, the central dichotomy that makes Theorem 2.4 and Theorem 1.2 go through is Theorem 3.5, which is introduced as "Now we can re-state [26, Theorem 1.10]", and Lemma 5.16 invokes the "almost Furstenberg decomposition [26]". Both citations are to a manuscript in preparation by the same authors and are not machine-checked, published, or otherwise independently verifiable; the paper's disclaimer ("We emphasise that the proof of that theorem in [26] is independent of [51], so no circularity is involved") addresses only circularity with respect to Stück–Zimmer, not the reliance on an inaccessible companion. Hence the central claim reduces to a load-bearing self-citation chain, giving score 6. One additional, non-circularity concern: in the proof of Theorem 2.4(b), η is set to (ε0/2C)^4 with ε0=ε0(Γ) from Lemma 3.6, and c_G is then defined via this η in (3.20); as written c_G depends on Γ, although the theorem states "c_G > 0 depends only on G". This is a correctness gap in the uniformity claim, not a circularity, since the fixed-Γ conclusion would still follow from the argument. It does not lower or raise the circularity score.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The ledger shows that the paper is conditional on the companion structure theory of [26]. No data-fitted constants or invented entities appear; all other inputs are standard results.

assumptions (4)
  • ad hoc to paper Theorem 3.5 (Almost Nevo-Zimmer dichotomy) is taken from [26, Theorem 1.10] as a black box
    Imported from the authors' unpublished companion manuscript; no proof or independent statement is given.
  • ad hoc to paper Almost Furstenberg decomposition used in Lemma 5.16 is taken from [26]
    Used to decompose Cesàro averages into P-almost invariant measures; proof not included in this preprint.
  • standard math Kazhdan property (T) and its L^1 form (Theorem 4.6 from [10])
    Higher-rank simple Lie groups have property (T); the L^1 version is a cited standard result.
  • domain assumption Borel density theorem / classification of ergodic invariant random subgroups ([2, Theorem 2.9])
    Used in Proposition 6.2 to reduce Stück-Zimmer to discrete invariant random subgroups.

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Cite this review

Pith. "Pith review of Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\"{u}ck-Zimmer Theorem." pith.science (2026). https://pith.science/paper/PHXRKUEP

@misc{pith2026260801382,
  author       = {Pith},
  title        = {Pith review of: Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\"uck-Zimmer Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PHXRKUEP}},
  note         = {Machine review of arXiv:2608.01382}
}
abstract

Fraczyk and Gelander proved in \cite{FG} that for any simple Lie group $G$ of high rank and for every non-lattice discrete subgroup $\Gamma\leq G$, the injectivity radius of points in $G/\Gamma$ is unbounded, resolving a conjecture of Margulis. In this work we obtain an explicit lower bound on the growth rate of the maximal injectivity radius of points taken from growing balls in $G/\Gamma$. More explicitly, we prove that for any $R>0$, one can embed a ball of radius $c\log^{(4)}R$ in $G/\Gamma$ centered at some point $[g]\in G/\Gamma$ where $g$ is taken from $G_R$ and for some constant $c=c(G,\Gamma)$. In particular, we show that for a general discrete subgroup $\Gamma$, if the injectivity radius growth in $G/\Gamma$ is slower than $\log^{(4)}$, $\Gamma$ must be a lattice. Additionally, we give a new, shorter and simpler proof of the Nevo-St\"{u}ck-Zimmer Theorem, saying that every action of a high rank simple group with property $(T)$ is either essentially free or essentially transitive. The results in this paper are obtained using the almost structure of measures from the accompanying paper, together with additional geometric considerations. As a step in the proof, we develop the following characterization for lattices. A discrete subgroup $\Gamma\leq G$ is a lattice if and only if there is a probability measure on $G/\Gamma$ which is sufficiently almost invariant under $G$. More precisely, suppose $\Gamma\leq G$ is a discrete subgroup for which there exists a probability measure $\nu$ on $G/\Gamma$ for which $W_1^{b}(g\nu,\nu)\leq \eps_0$ for some $\eps_0(\Gamma)>0$, then $\Gamma$ is a lattice.

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