Almost-invariant elements of group actions on Lip$_0$ spaces

Tom\'a\v{s} Raunig

REVIEW 3 major objections 5 minor 8 references

Almost-invariant elements of group actions on Lip$_0$ spaces

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves that on a free group with word metric, every almost-invariant Lipschitz function is within δ/2 of an invariant one, and the bound is sharp; finitely presented groups give the same conclusion with a constant depending only…

desk verdict A genuinely new positive result on Glasner's almost-invariant approximation problem in Lipschitz-free spaces; the main theorem is right, but the optimality proof needs a small repair. read the letter →

arxiv 2506.06770 v1 pith:TM66LYZK submitted 2025-06-07 math.FA

Tomáš Raunig This is my paper

classification math.FA MSC 46B2046B0420F65

keywords almost-invariantfunctionalsLipschitz-freespacesLip_0groupactionsbyisometriesfreegroupsfinitelypresentedwordmetricpartialquasimorphisms

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 asks when a Lipschitz function on a metric space that is almost unchanged by a group of isometries must be close to a function that is genuinely unchanged. It works in the dual of Lipschitz-free spaces, where such functions are elements of $\operatorname{Lip}_0(M)$. The main results show that in several natural settings the answer is yes: for free groups acting on themselves by translations every $\delta$-invariant function is within $\delta/2$ of an invariant one, and this constant cannot be improved; for finitely presented groups the same holds with a constant that depends only on the group. A further theorem covers metric spaces whose orbits shrink relative to the quotient metric. The paper also connects almost-invariance to partial quasimorphisms, recovering a known theorem on Lipschitz functions on uniformly discrete groups.

What carries the argument

The central object is the ratio-symmetric quantity $c_f(s,x) = \tfrac12(\sup_{g\in G}(f(gsx)-f(gx)) + \inf_{g\in G}(f(gsx)-f(gx)))$, which measures the mean growth of $f$ in the direction of $s$ at $x$. For a $\delta$-invariant $f$, this quantity is within $\delta/2 \cdot d(sx,x)$ of any individual increment $f(gsx)-f(gx)$, and it has the symmetry $c_f(s,x) = -c_f(s^{-1},x)$. On a group acting on itself by left translations, the map $g \mapsto c_f(g,e)$ is a homomorphism, hence an invariant element of $\operatorname{Lip}_0(G)$; the theorem for free groups states that this homomorphism is the best invariant approximation to $f$. The quotient transfer (Proposition 3.5) reduces the finitely presented case to finding a kernel-annihilating homomorphism close to $c_f$ on generators, which is solved by a finite-dimensional linear-algebra lemma.

What would settle it

The decisive check is scope: the paper's theorems apply only to actions on $\operatorname{Lip}_0(M)$ induced by isometries of $M$. Find an isometric action on some Lipschitz-free space $F(M)$ that does not arise this way — the paper cites [3] for their existence — and exhibit a $\delta$-invariant functional whose distance to the invariant subspace grows without bound relative to $\delta$. That would confirm the acknowledged limitation and show the positive results are tied to induced actions.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the almost-invariant functional approximation problem has a positive answer for Lipschitz-free spaces whenever the underlying metric geometry is sufficiently rigid. For a free group $F_S$ with word metric acting on itself by left translations, each $\delta$-invariant $f \in \operatorname{Lip}_0(F_S)$ has a closest invariant element $\bar{f}$, namely the homomorphism $\bar{f}(g) = \sum_{i=1}^n c_f(s_i,e)$ for a reduced word $g=s_1\cdots s_n$, and $\|f-\bar{f}\| \le \delta/2$; Example 3.3 shows the constant is sharp. The key identity is that invariant functions are exactly homomorphisms $G\to\mathbb{R}$, and the 'mean growth' $c_f(s,x) = \tfrac12(\sup_g (f(gsx)-f(gx)) + \inf_g (f(gsx)-f(gx)))$ is the quantity from which the best approximation is built. For finitely presented groups, the same idea — transported through the quotient map to a free group and solved by a finite-dimensional linear algebra lemma — gives a constant $C$ depending only on the presentation with $\|f-\bar{f}\| \le C\delta$. When no cyclic subgroup is bi-Lipschitz equivalent to $\mathbb{Z}$ and a fundamental domain controls the quotient metric, invariance is forced up to the constant $(2\alpha+1)\delta$.

Load-bearing premise

All main theorems assume the group action on $\operatorname{Lip}_0(M)$ is the one induced by an isometric action on the base pointed metric space $M$; the paper explicitly notes that not every isometric action on the Lipschitz-free space arises this way.

Editorial extensions

If this is right

For free groups with word-length metrics, the approximation constant $\delta/2$ is optimal, so any invariant approximation must be at least this close.
Finitely presented groups inherit the approximation with a constant depending only on the presentation; any failure of approximation, if it exists, must involve infinitely presented groups.
Under the fundamental-domain condition and the no-$\mathbb{Z}$-orbit assumption, the positive answer holds even for non-amenable groups, independently of amenability.
On groups with uniformly discrete invariant metrics, the almost-invariance result implies that every Lipschitz function is a partial quasimorphism with an explicit constant, recovering a known theorem.

Reading between the lines

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

A natural next step is to test whether the sharp $\delta/2$ bound for free groups extends to hyperbolic groups; the mean-growth homomorphism might serve as the canonical center there as well.
The proof of Theorem 3.7 computes the constant from a finite-dimensional relator matrix, so for any specific finitely presented group one could extract an explicit constant and test its sharpness computationally, which the paper does not do.
Because the theorems are confined to actions induced by base-space isometries, the natural boundary test is a non-induced isometric action on a Lipschitz-free space (known to exist by the cited work) on which some $\delta$-invariant functional lies far from every invariant one; that would mark the true scope of the positive results.

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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

Short version: this is a real contribution. The free-group theorem (Theorem 3.2), with the sharp delta/2 constant and the extremal example, is new. The finitely presented group theorem (Theorem 3.7) with a group-dependent constant C, and the non-Z-orbit structure theorem (Theorem 3.11), are also new sufficient conditions. Section 4's recovery of part of Kedra's theorem via the c_f machinery is a nice bonus. The averaging construction c_f = (sup + inf)/2 is the right tool, and the proofs of the main approximation statements are coherent.

Where I would push back: the optimality argument in Theorem 3.2 is not written correctly. It uses fbar(s) = c(s) for a general s in the group, which is only true for s in the generating set. The statement is very likely true: on a free group with the word metric, the Lipschitz norm is achieved on generator edges, so one can choose s in S, and then the displayed equality and the contradiction go through. But that step has to be added. This is a repairable gap, not a load-bearing flaw.

Other soft spots are cosmetic but real: internal cross-reference errors throughout (e.g., Corollary 2.2 called Theorem 2.2, Remark 2.5 called Theorem 2.5, Lemma 3.1 called Theorem 3.1), and the pseudometric case in Lemma 3.8 is terse and needs careful reading. The paper also explicitly limits itself to actions on Lip_0(M) induced by isometric actions on M; the wider class of isometric actions on the free space is left open, and the author acknowledges this.

On citations: the dependence on Glasner and Cuth-Doucha-Titkos is appropriate, and the relation to Kedra is clearly flagged. No red flags.

Bottom line: worth sending to a serious referee. With the missing generator-edge justification supplied and the cross-references fixed, I would be happy with it. A careful referee will likely find the same short list.

Referee Report

3 major / 5 minor

Summary. The paper studies Glasner's question of whether an almost-invariant functional on a Banach space can be approximated by an invariant one, in the setting where the Banach space is a Lipschitz-free space F(M) and the action is induced by an isometric action on the underlying pointed metric space. The main results are: (i) for a free group F_S with the word metric acting on itself by left translations, every delta-invariant f in Lip_0(F_S) admits an invariant approximant fbar with ||f - fbar|| <= delta/2, and fbar is claimed to be a best invariant approximation (Theorem 3.2); (ii) for finitely presented groups with word metrics, a positive answer holds with a group-dependent constant C (Theorem 3.7); (iii) for actions on metric spaces with a fundamental domain and no cyclic orbit bi-Lipschitz to Z, a positive answer holds with constant (2alpha+1)delta (Theorem 3.11). The paper also connects the notions to partial quasimorphisms and recovers a result of Kedra. The proofs are mostly explicit and constructive, using averaged increments c_f(s,x) and homomorphisms built from them.

Significance. If the results are correct, the paper gives the first positive answers for non-amenable groups in the Lipschitz-free space setting, and the sharp delta/2 constant for free groups is a particularly clean and strong statement. The methods, based on the averaged increments c_f and the construction of invariant homomorphisms, are natural and likely to be useful in further work on almost-invariant functionals. The paper also explicitly acknowledges the limitation that not every isometric action on a Lipschitz-free space arises from an action on the underlying metric space, which is appropriate. The connection to partial quasimorphisms in Section 4 is a useful remark, but that section currently contains a false theorem as stated.

major comments (3)

[Theorem 3.2 (optimality part)] The proof of the 'moreover' claim asserts the equality |f(xs)-f(x)-c(s)|/d(s,e) = |f(xs)-f(x)-fbar(s)|/d(s,e) for arbitrary x,s in G. This is not generally true because fbar(s) is defined as the sum of c(s_i) over the reduced word representing s, so fbar(s)=c(s) only when s lies in the generating set S (or when the c-values are additive along reduced words). The contradiction is therefore not established as written. The gap is repairable: on the Cayley graph of a free group with the word metric, every phi in Lip_0 satisfies ||phi|| = max_{x in G, s in S} |phi(xs)-phi(x)|, since any two points are joined by a geodesic path of unit-length generator edges. Thus the chosen x,s may be taken with s in S, and then fbar(s)=c(s) and the argument goes through. This edge characterization should be stated and used explicitly.
[Lemma 3.6 (used in Theorem 3.7)] The proof of Lemma 3.6 contains an invalid compactness step. The elements u_epsilon are chosen in ker A, a subspace of X, and the proof appeals to finite-dimensionality of \tilde Y to conclude that a ball is compact; but the ball in ker A (or in X) need not be compact. The lemma itself is true and standard: since Im A is finite-dimensional, one can choose a bounded linear right-inverse S: Im A -> X with A S = Id, and then u = x - S A x lies in ker A with ||x-u|| <= ||S|| ||Ax||. The proof should be replaced by this argument; as written, the proof of Theorem 3.7 is incomplete because it relies on Lemma 3.6.
[Theorem 4.3] The statement is false as written. The proof shows that f_e = f - f(e) is a partial quasimorphism, but the final step tries to bound |f(e)| by (|f(e)|/A) min{d(g,e), d(h,e)}, which fails when g=e or h=e. Since Definition 4.1 forces any partial quasimorphism to vanish at the identity, the theorem would imply f(e)=0 for every Lipschitz f on a uniformly discrete group, which is false (e.g., the constant function 1 on Z). The theorem should be corrected to state that f_e is a partial quasimorphism, or to assume f(e)=0.

minor comments (5)

[Throughout] There are numerous internal cross-reference errors: 'Theorem 2.2' should be 'Corollary 2.2' (proof of Corollary 2.3), 'Theorem 2.5' should be 'Remark 2.5' (proof of Theorem 4.3), 'Theorem 3.1' should be 'Lemma 3.1' (proof of Theorem 3.2), and 'Theorem 2.1' should be 'Lemma 2.1' in several places (e.g., Proposition 2.7, Proposition 3.5, Lemma 3.9, Theorem 4.2). These should be corrected throughout.
[Theorem 3.11] In condition (i), the phrase 'for every pair g in G, x in X' uses the symbol X, which is not defined; it should be 'x in M'.
[Abstract and Introduction] The word 'underlaying' should be 'underlying'.
[Theorem 4.3 proof] The line 'Let A > 0 be such that for any g in G we have A <= d(g,e)' is impossible for g=e; it should say 'for every g != e'. This is related to the major issue in Theorem 4.3, but the phrasing should be fixed even if the statement is corrected.
[Lemma 3.8 proof] The displayed inequality in the final case contains the notation 'k m' and 'kmk', which is confusing and likely a typesetting artifact; please clarify the intended exponents or factorizations.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the invariant approximants are explicitly constructed from the given function f and then proved to approximate it; no fitted input is renamed as a prediction, and no load-bearing self-citation is used.

full rationale

The paper's central results are not circular. Theorem 3.2 defines c(s) = c_f(s,e) as the midpoint of sup_g(f(gs)-f(g)) and inf_g(f(gs)-f(g)), and defines fbar(g) as the sum of c(s_i) over the reduced word spelling g. This is an explicit construction from the given almost-invariant f, not an input equivalent to the target. The theorem then proves fbar is a homomorphism and hence invariant by Corollary 2.3, and proves the bound ||f-fbar|| <= delta/2 using Proposition 2.6, which bounds each increment in terms of delta-invariance. The optimality argument compares fbar with an arbitrary invariant approximant using the same c(s), so the conclusion is not assumed as a premise. Theorem 3.7 transfers the free-group construction to finitely presented groups via a finite-rank linear operator and Lemma 3.6, with the constant depending only on the group presentation; it is not fitted to any particular f or to the target bound. Section 4 recovers an implication from Kedra's theorem as an independent external result, not as an input. The one citation to Cuth-Doucha-Titkos is used only to acknowledge that not every isometric action on a Lipschitz-free space arises from an isometric action on the underlying pointed space; this is an honest scope limitation, not a load-bearing premise. Even if a referee finds a gap in the optimality proof of Theorem 3.2 (for example, the equality involving fbar(s) for non-generators), that is a correctness issue, not a circular reduction of the theorem to its own assumptions.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claims rest on the standard duality F(M)*=Lip_0(M) and on the restriction to actions induced from the metric space. No free parameters are fitted to data, and no new physical or mathematical entities are postulated; the averaging quantity c_f is constructed from the function being approximated.

assumptions (2)

standard math The dual of the Lipschitz-free space F(M) is Lip_0(M) with the Lipschitz norm (Weaver [7]).
Invoked in Section 1 to reformulate almost-invariant functionals on F(M) as almost-invariant functions in Lip_0(M).
domain assumption The action on Lip_0(M) is the one induced by an isometric action on the underlying pointed metric space, alpha(g,f)(x)=f(g^-1 x)-f(g^-1 0).
All main theorems are stated for actions of this form; the paper notes not every isometric action on a Lipschitz-free space arises this way, citing Cuth-Doucha-Titkos [3].

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Cite this review

Pith. "Pith review of Almost-invariant elements of group actions on Lip$_0$ spaces." pith.science (2026). https://pith.science/paper/TM66LYZK

@misc{pith2026250606770,
  author       = {Pith},
  title        = {Pith review of: Almost-invariant elements of group actions on Lip$_0$ spaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TM66LYZK}},
  note         = {Machine review of arXiv:2506.06770}
}

read the original abstract

This work is motivated by a question published in E. Glasner's paper On a question of Kazhdan and Yom Din regarding the possibility to approximate functionals on a Banach space which are almost invariant with respect to an action of a discrete group by functionals that are invariant. We study the case when the Banach space is a Lipschitz-free space equipped with an action induced by an action by isometries on the underlaying space. We find a few different conditions sufficient for the answer to be positive; for example the case of free or finitely presented groups endowed with left-invariant metrics acting on themselves by translations. Relations to quasimorphisms are also briefly studied.

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Reference graph

Works this paper leans on

8 extracted references · 7 canonical work pages

[1]

M. B. Bekka, P. d. La Harpe, and A. Valette.Kazhdan ’s property (T). New mathematical monographs

work page
[2]

Calegari.scl, volume 20 ofMSJ Memoirs

D. Calegari.scl, volume 20 ofMSJ Memoirs. Mathematical Society of Japan, Tokyo, 2009

work page 2009
[3]

C´ uth, M

M. C´ uth, M. Doucha, and T. Titkos. Isometries of Lipschitz-free banach spaces.Journal of the London Mathematical Society, 110(5):e70000, 2024

work page 2024
[4]

E. Glasner. On a question of Kazhdan and Yom Din.Israel Journal of Mathematics, 251(2):467–493, Dec 2022

work page 2022
[5]

J. K¸ edra. On Lipschitz functions on groups equipped with conjugation-invariant norms.Colloquium Math- ematicum, 174(1):89–99, Oct. 2023

work page 2023
[6]

Quarteroni, R

A. Quarteroni, R. Sacco, and F. Saleri.Numerical Mathematics. Springer New York, 2007

work page 2007
[7]

Weaver.Lipschitz Algebras

N. Weaver.Lipschitz Algebras. WORLD SCIENTIFIC, Singapore, 2nd edition, 2018

work page 2018
[11]

Cambridge University Press, Cambridge, 2008

2008

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

M. B. Bekka, P. d. La Harpe, and A. Valette.Kazhdan ’s property (T). New mathematical monographs

work page

[2] [2]

Calegari.scl, volume 20 ofMSJ Memoirs

D. Calegari.scl, volume 20 ofMSJ Memoirs. Mathematical Society of Japan, Tokyo, 2009

work page 2009

[3] [3]

C´ uth, M

M. C´ uth, M. Doucha, and T. Titkos. Isometries of Lipschitz-free banach spaces.Journal of the London Mathematical Society, 110(5):e70000, 2024

work page 2024

[4] [4]

E. Glasner. On a question of Kazhdan and Yom Din.Israel Journal of Mathematics, 251(2):467–493, Dec 2022

work page 2022

[5] [5]

J. K¸ edra. On Lipschitz functions on groups equipped with conjugation-invariant norms.Colloquium Math- ematicum, 174(1):89–99, Oct. 2023

work page 2023

[6] [6]

Quarteroni, R

A. Quarteroni, R. Sacco, and F. Saleri.Numerical Mathematics. Springer New York, 2007

work page 2007

[7] [7]

Weaver.Lipschitz Algebras

N. Weaver.Lipschitz Algebras. WORLD SCIENTIFIC, Singapore, 2nd edition, 2018

work page 2018

[8] [11]

Cambridge University Press, Cambridge, 2008

2008