{"id":"58647978-5dd2-4ca5-9854-c980d311111c","arxiv_id":"1908.10216","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors fix their Euclidean factor theorem for CAT(0) lattices: a lattice always commensurates a free abelian subgroup whose rank equals the Euclidean dimension, and finite generation or residual finiteness recovers the original splitting.","lead":"This erratum corrects a false theorem from the authors' 2009 paper on isometry groups of non-positively curved spaces. It replaces the incorrect virtual splitting of a lattice by a Z^n factor with a correct statement about commensurated free abelian subgroups, and it extends the result to non-finitely generated lattices.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The converse half of Theorem 2(i) invokes Proposition 4 for a subgroup only known to be commensurated by the lattice Γ, not by the ambient group; the proof as written does not bridge this gap.","rationale":"The reader identified the canonical decomposition of Is(X) as the weakest assumption, and that is indeed an external but standard ingredient. However, the more specific load-bearing gap in the written proof is the application of Proposition 4. That proposition has precise hypotheses about commensurability by a minimally acting ambient group and finite-dimensional Tits boundary. In the proof of Theorem 2(i), the subgroup H is commensurated by the lattice Γ, not by the ambient group G, and no argument is given to pass to the closure of the projection of Γ to Is(Y) or to verify that this closure acts minimally on the relevant irreducible factors. The erratum's central claim is likely true and the gap is probably fillable using results such as [CM09b, Th. 3.14], but as written the proof does not establish the hypothesis. Therefore the verdict should be CONDITIONAL rather than unconditional ACCEPT: the paper should be accepted once this step is explicitly justified or a reference is supplied.","tokens_in":10722,"tokens_out":31059,"duration_ms":344914,"concrete_test":"Verify the missing bridge in the proof of Theorem 2(i): take a concrete CAT(0) lattice Γ, for instance the one from Example 7 or the Leary–Minasyan example, and compute the convex limit set Δ_H of a commensurated abelian subgroup H. Check whether Δ_H is invariant under the full isometry group of each irreducible non-Euclidean factor or only under the projection of Γ. Equivalently, determine whether the closure of the projection of Γ to D = Is(Y) acts minimally on Y; if it leaves a proper closed convex invariant subset, Proposition 4 cannot be invoked as written and the proof of the m ≤ n bound requires an additional argument.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The converse part of Theorem 2(i) needs to show that every commensurated abelian subgroup H of Γ acts properly on the Euclidean factor E, which forces m ≤ n. The proof says: 'Therefore, Proposition 4 implies that H fixes a point in each irreducible factor of Y'. But Proposition 4 requires the subgroup to be commensurated by the ambient group G acting minimally and without a fixed point at infinity on an irreducible space with finite-dimensional Tits boundary. In the present situation H is only known to be commensurated by the lattice Γ, not by the canonical group Is(Y), and the text does not justify that the closure of the projection of Γ to Is(Y) acts minimally on each irreducible factor of Y, nor that commensurability by Γ passes to commensurability by that closure. This is not a challenge to the cited decomposition of Is(X) itself, but to the internal application of Proposition 4. Since the entire converse direction of part (i) depends on this step, the proof of Theorem 2(i) is incomplete as written unless the missing bridge is supplied.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This erratum corrects Theorem 1.3(i) of the authors' 2009 paper [CM09b]. It identifies the error as the assertion in the proof of Proposition 3.6 that the commensurator of any lattice in a Euclidean motion group is virtually abelian, and replaces Theorem 1.3(i) by Theorem 2. The new theorem states that if X is a proper CAT(0) space with maximal Euclidean factor R^n, G < Is(X) is closed, acts minimally and cocompactly, and Γ < G is any lattice, then Γ commensurates a free abelian subgroup isomorphic to Z^n, n is the largest such rank, and every commensurated abelian subgroup of Γ acts properly on R^n. With finite generation, the paper recovers virtual splitting results under suitable extra hypotheses. The proof uses a canonical decomposition Is(X) = S × A × D, a new Proposition 4 on commensurated subgroups of irreducible CAT(0) spaces, Lemma 5 on normalizers in Euclidean motion groups, and the Leary–Minasyan construction as a counterexample. The final section lists the dependent corrections to the original paper, including a repaired proof of Theorem 4.2 and a corrected statement of Theorem 6.1.","tokens_in":10856,"tokens_out":11661,"duration_ms":135611,"significance":"If correct, this paper repairs a false theorem in a standard reference and identifies the correct formulation, replacing 'free abelian normal subgroup' by 'commensurated free abelian subgroup' and removing the finite generation hypothesis in the main existence statement. The authors are transparent about the error, explicitly locate the false assertion, provide a concrete counterexample, and supply detailed corrections to the dependent results. The paper also includes a new example showing that finite generation cannot be dropped in Theorem 2(ii). The overall framework is standard and I see no circularity. However, the converse direction of the central new theorem currently contains a proof gap that needs to be repaired before the paper can be accepted.","major_comments":[{"comment":"The sentence 'Therefore, Proposition 4 implies that H fixes a point in each irreducible factor of Y' is not justified by the hypotheses as written. Proposition 4 applies to a subgroup H0 of a group G0 acting minimally on an irreducible space, with H0 commensurated by G0. In the proof, H is only known to be commensurated by the lattice Γ, not by the ambient group acting on Y. The natural ambient group on the Y-factor would be the closure of the projection of Γ to Is(Y), or its projection to each irreducible factor, and the proof does not establish either (a) that commensurability by Γ passes to commensurability by that closure, or (b) that this closure acts minimally on each irreducible factor of Y (or otherwise satisfies the hypotheses of Proposition 4). This step is load-bearing: it is exactly what forces the closure of the projection of H to S × D to be compact and hence makes the H-action on the Euclidean factor proper. The converse half of Theorem 2(i) is therefore incomplete as written. On my reading this is a local gap that can be repaired by a supplementary lemma, not a disproof of the theorem, but it must be supplied.","section":"§1, proof of Theorem 2(i)"}],"minor_comments":[{"comment":"The proof passes from finite index subgroups to commensurable subgroups rather quickly: the sentence 'If H0 < H is a finite index subgroup...' establishes invariance of the convex limit set under finite index replacement, and the extension to commensurable subgroups is stated without proof. The step is immediate from the definition, but it could be spelled out for clarity.","section":"§1, Proposition 4"},{"comment":"There are small typographical issues in this example: 'the the Prüfer 2-group' contains a duplicated article, and the notation B = Z[1/2]/Z should make explicit that the quotient is taken in the category of abelian groups, not rings.","section":"§1, Example 7"},{"comment":"The text contains the typo 'throuhout' in the phrase 'throughout the Introduction'. This should be corrected.","section":"§2, Further corrections"},{"comment":"The reference [CKRW] is listed as 'preprint, to appear' without an arXiv identifier. Adding the arXiv number would be helpful to readers.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The main proof gap in Theorem 2(i) is local and seems fixable: the authors need to justify that the commensurability of H by Γ passes to the closure of the projection of Γ to the Y-factor, and that this closure acts minimally on each irreducible factor. I would be happy to see a revised version with this missing bridge supplied. The paper is otherwise a responsible and useful erratum."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper is exactly what an erratum should be: it identifies a false theorem from the authors' 2009 paper, says what the actual mistake was, and supplies a corrected statement that is genuinely different and more subtle. The new Theorem 2(i), with 'commensurated' in place of 'normal', and the extension to lattices that are not finitely generated, are real improvements. The debt to Leary-Minasyan is acknowledged directly, and the paper goes through the earlier results one by one, telling the reader what survives and what needs modification. That is good scholarly hygiene.\n\nThe proofs are mostly careful and use standard tools. I believe the main assertions are likely correct. But the stress-test concern about the converse of Theorem 2(i) deserves a referee's attention. The proof says that Proposition 4 implies that a commensurated abelian subgroup H fixes a point in each irreducible factor of Y. Proposition 4 requires H to be commensurated by the ambient group acting minimally on that factor. In the present situation, H is only known to be commensurated by the lattice Γ, not by the closure of Γ's projection to the isometry group of Y (or of each irreducible factor). The proof does not explain why commensurability by Γ passes to commensurability by that closure, or why that closure acts minimally. This is not a fatal flaw in the architecture of the erratum, but it is a gap in the proof as written. If it cannot be filled, the converse direction of part (i) needs a different argument.\n\nAlso worth noting: the paper leans on the canonical decomposition from the authors' earlier structure paper. That is fine for an erratum, but it means the corrected theorem inherits any issues in that decomposition.\n\nWho is this for? Anyone who cited or used the original Theorem 1.3(i) needs to read this. Geometric group theorists studying CAT(0) lattices should have it on the record. It deserves peer review; the authors should be asked to close the gap around Proposition 4.","headline":"A candid erratum that fixes a real false theorem, with one proof step that looks under-justified.","tokens_in":11442,"tokens_out":3860,"would_cite":true,"duration_ms":37699,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","22D05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A corrected Euclidean-factor theorem for CAT(0) lattices replaces a false virtual-splitting claim with a commensurability statement.","keywords":["CAT(0) spaces","lattices in isometry groups","Euclidean factor theorem","commensurated subgroups","virtual splitting","non-positive curvature","locally compact groups"],"falsifier":"Apply Theorem 2(i) to the lattice $\\Gamma < (\\mathbb{R}^2 \\rtimes SO(2)) \\times D$ built from the Pythagorean rotation in the paper's Example 1: the theorem predicts $\\Gamma$ commensurates a free abelian subgroup whose projection to the Euclidean factor is a lattice of rank 2. Directly computing this commensurated subgroup from the semidirect product $\\Lambda = (\\mathbb{Z}[1/5])^2 \\rtimes_\\alpha \\mathbb{Z}$ is a concrete check; failing to find it would refute the corrected theorem.","tokens_in":2354,"feed_emoji":"📐","tokens_out":2230,"duration_ms":102722,"temperature":0.7,"pith_summary":"This erratum corrects a false statement from the 2009 paper: a CAT(0) lattice need not virtually split as $\\mathbb{Z}^n \\times \\Gamma'$, where $n$ is the dimension of the maximal Euclidean factor. The counterexample, built from a Pythagorean rotation, embeds as a cocompact lattice in the product of a Euclidean motion group and a group acting on a tree; it has no finite-index free abelian normal subgroup of rank 2, yet it still commensurates a copy of $\\mathbb{Z}^2$. The replacement, Theorem 2(i), states that for any proper CAT(0) space $X$ with maximal Euclidean factor $\\mathbb{R}^n$, any lattice $\\Gamma$ in a closed minimal cocompact subgroup of $\\operatorname{Is}(X)$ commensurates a free abelian subgroup $\\Gamma_A \\cong \\mathbb{Z}^n$, that $n$ is the largest such rank, and that every commensurated abelian subgroup of $\\Gamma$ acts properly on $\\mathbb{R}^n$. No finite-generation assumption is needed for this part; the original virtual splitting is recovered when $\\Gamma$ is finitely generated and residually finite. The paper then lists the minor changes to the dependent theorems and shows by another example that the finite-generation hypothesis cannot be dropped in Theorem 2(ii).","feed_headline":"Corrected theorem: CAT(0) lattices commensurate a Z^n subgroup","feed_subtitle":"The false 2009 virtual-splitting claim is replaced by commensurability; splitting returns for finitely generated residually finite lattices.","key_machinery":"The machinery is the canonical decomposition of $\\operatorname{Is}(X)$ into $S \\times A \\times D$ — semisimple, Euclidean motion, totally disconnected — together with the shift from normal to commensurated subgroups. A subgroup is commensurated when each conjugate intersects it in finite index both ways; this is exactly the invariance property needed for the new statement. Proposition 4 carries the geometric weight: for an irreducible proper CAT(0) space with finite-dimensional Tits boundary, a commensurated subgroup either fixes a point or still acts minimally, using the convex limit set and a circumradius argument. Lemma 5 adds the Euclidean-side fact that the normalizer of any lattice in $\\mathbb{R}^k \\rtimes O(k)$ is virtually contained in the translation group. Together these turn commensurability of the abelian subgroup into properness of its action on the Euclidean factor, and the rank bound follows.","core_discovery":"The central claim is Theorem 2(i), the amended Euclidean Factor Theorem. For a proper CAT(0) space $X$ whose maximal Euclidean factor is $E \\cong \\mathbb{R}^n$, a closed subgroup $G < \\operatorname{Is}(X)$ acting minimally and cocompactly, and any lattice $\\Gamma < G$, the theorem asserts that $\\Gamma$ commensurates a free abelian subgroup $\\Gamma_A \\cong \\mathbb{Z}^n$, that $n$ is the largest rank of any free abelian commensurated subgroup, and that any commensurated abelian subgroup of $\\Gamma$ acts properly on $E$. The proof decomposes the full isometry group as $\\operatorname{Is}(X) = S \\times A \\times D$, with $S$ a semisimple Lie group with trivial centre and no compact factors, $A \\cong \\mathbb{R}^n \\rtimes O(n)$ the Euclidean motion group, and $D$ totally disconnected; this comes from the earlier structure theory. A lattice lemma shows that some finite-index commensurated subgroup intersects $D$ in a compact open subgroup and, after passing through the Euclidean factor, produces the free abelian $\\Gamma_A$ of rank $n$. The converse direction uses a new proposition: a commensurated subgroup of a minimal action on an irreducible non-Euclidean CAT(0) space either fixes a point or still acts minimally, so an abelian commensurated subgroup cannot act minimally on the non-Euclidean part and therefore must act properly on the Euclidean factor. The paper also proves that if $\\Gamma$ is finitely generated and residually finite, the original virtual splitting into $\\mathbb{Z}^n \\times \\Gamma'$ does hold, via the profinite closure of $\\Gamma_A$.","pith_inferences":["A natural test is whether Theorem 2(i) persists for non-uniform lattices in more general locally compact groups, where no CAT(0) space is present; the statement only uses the commensurated lattice in $S \\times A \\times D$, so it may extend to a purely locally-compact-group formulation.","The erratum suggests a systematic translation principle: wherever the original proofs used a normal free abelian subgroup of rank $n$, the same conclusion may hold with 'normal' replaced by 'commensurated' whenever the lattice is not residually finite; checking the remaining dependent theorems against this principle could reveal further corrections.","One could ask whether the rank $n$ in Theorem 2(i) is a genuine commensurability invariant of the lattice, independent of the ambient CAT(0) space; if so, the Euclidean factor dimension would be determined by the abstract commensurability class of $\\Gamma$, a point the paper does not address."],"forward_implications":["Theorem 1.3(i) of the original paper is replaced by Theorem 2, and all other results in the original paper remain valid with the listed corrections.","Corollary 3: the amenable radical of any CAT(0) lattice is virtually $\\mathbb{Z}^k$ for some $k \\leq n$; for finitely generated lattices there is a finite-index subgroup splitting as $\\mathbb{Z}^k \\times \\Gamma'$ with $\\Gamma'$ having trivial amenable radical, together with a corresponding splitting of the space.","Theorem 2(iv) restores the original virtual splitting $\\mathbb{Z}^n \\times \\Gamma'$ for finitely generated residually finite lattices.","The hypothesis in Theorem 4.11 is changed from 'do not virtually split a $\\mathbb{Z}^n$ factor' to 'do not commensurate a $\\mathbb{Z}^n$ subgroup', and Corollary 4.13 holds without change.","Theorem 6.1 is corrected: the normal subgroup $\\Gamma_D$ is either finite or infinitely generated, and the quotient is an arithmetic lattice in a product of semisimple Lie and algebraic groups; the finite-generation hypothesis cannot be dropped in Theorem 2(ii), as Example 7 shows."],"supporting_citations":[{"why":"Contains the false Theorem 1.3(i) that this paper corrects, together with the lemmas (3.2, 3.4, 3.5, 3.14, 3.15) reused in the new proofs.","marker":"[CM09b]"},{"why":"Supplies Theorem 1.6 and Addendum 1.8 giving the canonical $\\operatorname{Is}(X)$-invariant decomposition $X \\cong M \\times \\mathbb{R}^n \\times Y$, and Theorem 1.10 constraining normal abelian subgroups.","marker":"[CM09a]"},{"why":"Theorem M is the fixed-point-at-infinity input that lets the proof pass from $G$ to the full isometry group and obtain the decomposition.","marker":"[CM13]"},{"why":"Provides the CAT(0) lattice counterexample, a semidirect product built from a Pythagorean rotation, that disproves the original Theorem 1.3(i).","marker":"[LM]"},{"why":"Yields the profinite closure result used to show that a finitely generated residually finite lattice virtually splits.","marker":"[CKRW]"},{"why":"Supplies Proposition 1.4 on circumradius, used in Proposition 4 to rule out a fixed point at infinity.","marker":"[BL05]"},{"why":"Provides the Cayley–Abels graph construction used in the counterexample to build the totally disconnected factor acting on a tree.","marker":"[Abe74]"},{"why":"Records Auslander's theorem for connected Lie groups, the contrast point for the new example's behaviour modulo the amenable radical.","marker":"[Rag72]"}],"fun_headline_variants":["Corrected CAT(0) factor theorem: commensurate Z^n for all lattices","Euclidean factor theorem amended: Z^n always commensurated in lattice","Non-finitely generated lattices included in corrected CAT(0) theorem","Splitting restored for finitely generated residually finite lattices","CAT(0) lattice theorem fix: rank n free abelian always, splitting if residually finite"],"cache_read_input_tokens":13568,"weakest_assumption_plain":"The proof depends on the previously established structure theorem that the full isometry group of a proper CAT(0) space with a minimal cocompact closed subgroup decomposes canonically as $S \\times A \\times D$ with $S$ semisimple, $A$ Euclidean, and $D$ totally disconnected; if that decomposition fails in some such space, the corrected Euclidean-factor theorem collapses.","fun_headline_variants_meta":{"raw":{"variants":["Corrected CAT(0) factor theorem: commensurate Z^n for all lattices","Euclidean factor theorem amended: Z^n always commensurated in lattice","Non-finitely generated lattices included in corrected CAT(0) theorem","Splitting restored for finitely generated residually finite lattices","CAT(0) lattice theorem fix: rank n free abelian always, splitting if residually finite"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001395,"raw_usage":{"total_tokens":5644,"prompt_tokens":948,"completion_tokens":4696,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":4592}},"tokens_in":564,"tokens_out":4696,"duration_ms":32494,"temperature":1.0,"reasoning_tokens":4592,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:04:16.674374+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Apply Theorem 2(i) to the lattice $\\Gamma < (\\mathbb{R}^2 \\rtimes SO(2)) \\times D$ built from the Pythagorean rotation in the paper's Example 1: the theorem predicts $\\Gamma$ commensurates a free abelian subgroup whose projection to the Euclidean factor is a lattice of rank 2. Directly computing this commensurated subgroup from the semidirect product $\\Lambda = (\\mathbb{Z}[1/5])^2 \\rtimes_\\alpha \\mathbb{Z}$ is a concrete check; failing to find it would refute the corrected theorem.","supporting_citations":[],"review_version":1}