{"id":"31c62eeb-d3c4-4669-a8ee-018434d3c0d5","arxiv_id":"2506.06770","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Almost-invariant Lipschitz functions on free groups are within delta/2 of an invariant homomorphism, and finitely presented groups admit group-dependent linear approximation.","lead":"This paper proves that almost-invariant Lipschitz functions on free groups are approximated by invariant ones with the best possible error. The same approach gives positive answers for finitely presented groups and for metric spaces without integer-like cyclic orbits.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Best-approximation proof in Theorem 3.2 uses fbar(s)=c(s) for non-generators, which is false in general; with s restricted to the generating set the argument can be repaired, but as written the optimality claim is unproved.","rationale":"I read the paper in good faith. The main construction is a coherent projection onto invariant homomorphisms by averaging edge cocycles, and the delta/2 bound follows from the edge-range estimate; the finitely presented extension via the quotient and Lemma 3.6 also appears coherent. The reader's weakest assumption was the restriction to induced actions, which the paper explicitly acknowledges and which does not threaten Theorem 3.2 within its stated assumptions. The most load-bearing concern I found is internal: the proof of the best-approximation part of Theorem 3.2 conflates c(s) with fbar(s) for arbitrary group elements. This is a genuine proof gap, but it is localized and repairable by restricting to generators, so the overall conditional verdict is unchanged. The central delta/2 approximation claim is not affected by this gap.","tokens_in":16033,"tokens_out":37257,"duration_ms":388759,"concrete_test":"Re-derive the optimality step of Theorem 3.2 with s restricted to the generator set S, using the equality ||phi|| = max_{x in G, s in S} |phi(xs)-phi(x)| for word-metric Cayley graphs. Check that the contradiction chain c^+(s)-c^-(s) <= 2 sup_g |f(gs)-f(g)-ef(s)| < 2|f(xs)-f(x)-c(s)| <= c^+(s)-c^-(s) works for a generator s. Also verify the unit-step example on Z: with f(n)=0 for n<=0 and f(n)=1 for n>0, delta=1, c(2)=1/2 but fbar(2)=1, confirming the original equality is false for non-generators. If the generator-restricted proof is valid, the theorem stands; if not, search for a homomorphism closer to f than fbar.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In the optimality part of Theorem 3.2 the proof 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 an identity: fbar(g) is defined as the sum of c(s_i) over the reduced word of g, so fbar(s)=c(s) only when s lies in the generating set S or when the c-values happen to be additive along reduced words. For the delta-invariant unit-step function on Z (f(n)=0 for n<=0, f(n)=1 for n>0, delta=1), one has c(2)=1/2 while fbar(2)=1, so the asserted equality fails for s=2. Consequently the contradiction proving the 'best invariant approximation' claim is not established as written. The gap is repairable: on the Cayley graph of a free group with word metric, every phi in Lip_0 satisfies ||phi|| = max_{x in G, s in S} |phi(xs)-phi(x)|, because 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; then fbar(s)=c(s) and the rest of the argument goes through. The paper does not state or use this edge characterization, so the proof is incomplete even though the theorem is likely true.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16296,"tokens_out":16201,"duration_ms":145889,"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":[{"comment":"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.","section":"Theorem 3.2 (optimality part)"},{"comment":"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.","section":"Lemma 3.6 (used in Theorem 3.7)"},{"comment":"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.","section":"Theorem 4.3"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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'.","section":"Theorem 3.11"},{"comment":"The word 'underlaying' should be 'underlying'.","section":"Abstract and Introduction"},{"comment":"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.","section":"Theorem 4.3 proof"},{"comment":"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.","section":"Lemma 3.8 proof"}],"recommendation":"major_revision","confidential_remarks":"The paper contains several promising and likely correct results, and the core ideas are sound. However, the proof of the optimality claim in Theorem 3.2 has a nontrivial gap, the proof of Lemma 3.6 is invalid (though the lemma is true), and Theorem 4.3 is false as stated. These issues are repairable within the manuscript's scope, so I recommend major revision rather than rejection. The number and type of cross-reference errors also suggest that the manuscript needs a careful proofreading pass."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhere 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.\n\nOther 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.\n\nOn citations: the dependence on Glasner and Cuth-Doucha-Titkos is appropriate, and the relation to Kedra is clearly flagged. No red flags.\n\nBottom 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.","headline":"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.","tokens_in":16849,"tokens_out":5554,"would_cite":true,"duration_ms":54400,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46B20","46B04","20F65"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["almost-invariant functionals","Lipschitz-free spaces","Lip_0 spaces","group actions by isometries","free groups","finitely presented groups","word metric","partial quasimorphisms"],"falsifier":"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.","tokens_in":15806,"feed_emoji":"📏","tokens_out":12367,"duration_ms":97527,"temperature":0.7,"pith_summary":"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.","feed_headline":"Almost-invariant functions on free groups hit δ/2 bound","feed_subtitle":"Proves the sharp δ/2 bound for free groups and a uniform, group-dependent bound for finitely presented groups.","key_machinery":"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.","core_discovery":"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 $(2\\alpha+1)\\delta$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the motivating almost-invariant functional question and the known positive cases that this paper extends.","marker":"[4]"},{"why":"Provides the standard identification $F(M)^* = \\operatorname{Lip}_0(M)$, the Lipschitz norm, and the linearization property used to define the induced actions.","marker":"[7]"},{"why":"Shows not every isometric action on a Lipschitz-free space is induced by a base-space isometry; the paper notes this scope limitation.","marker":"[3]"},{"why":"Gives the theorem on Lipschitz functions being partial quasimorphisms, which Section 4 recovers as an application.","marker":"[5]"},{"why":"Supplies the standard reference for quasimorphisms, used in the definitions and context of Section 4.","marker":"[2]"}],"fun_headline_variants":["Sharp δ/2 bound for almost-invariant functions on free groups","Free groups: almost-invariant functionals get invariant within δ/2","Approximating almost-invariant functionals: sharp δ/2 for free groups","Lipschitz-free spaces: δ-invariant functions have closest invariant at δ/2"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Sharp δ/2 bound for almost-invariant functions on free groups","Free groups: almost-invariant functionals get invariant within δ/2","Approximating almost-invariant functionals: sharp δ/2 for free groups","Lipschitz-free spaces: δ-invariant functions have closest invariant at δ/2"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000296,"raw_usage":{"total_tokens":1730,"prompt_tokens":972,"completion_tokens":758,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":674}},"tokens_in":588,"tokens_out":758,"duration_ms":6593,"temperature":1.0,"reasoning_tokens":674,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:51:27.636091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the motivating almost-invariant functional question and the known positive cases that this paper extends."},{"cited_title":"Weaver.Lipschitz Algebras","cited_arxiv_id":null,"evidence_quote":"Provides the standard identification $F(M)^* = \\operatorname{Lip}_0(M)$, the Lipschitz norm, and the linearization property used to define the induced actions."},{"cited_title":"C´ uth, M","cited_arxiv_id":null,"evidence_quote":"Shows not every isometric action on a Lipschitz-free space is induced by a base-space isometry; the paper notes this scope limitation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the theorem on Lipschitz functions being partial quasimorphisms, which Section 4 recovers as an application."},{"cited_title":"Calegari.scl, volume 20 ofMSJ Memoirs","cited_arxiv_id":null,"evidence_quote":"Supplies the standard reference for quasimorphisms, used in the definitions and context of Section 4."}],"review_version":1}