REVIEW 1 major objections 4 minor 14 references
Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups"
T0 review · 1 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A corrected Euclidean-factor theorem for CAT(0) lattices replaces a false virtual-splitting claim with a commensurability statement.
desk verdict A candid erratum that fixes a real false theorem, with one proof step that looks under-justified. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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.
What would settle it
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.
Extended reading notes
Core claim
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 $\Gamma_A$.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (1)
- [§1, proof of Theorem 2(i)] 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.
minor comments (4)
- [§1, Proposition 4] 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.
- [§1, Example 7] 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.
- [§2, Further corrections] The text contains the typo 'throuhout' in the phrase 'throughout the Introduction'. This should be corrected.
- [References] The reference [CKRW] is listed as 'preprint, to appear' without an arXiv identifier. Adding the arXiv number would be helpful to readers.
Circularity Check
No circularity: the erratum corrects the authors' own false theorem using an external counterexample and independent structural inputs.
full rationale
The paper's central Theorem 2 is not derived from its own conclusion. It begins by acknowledging that Theorem 1.3(i) of [CM09b] is false, citing the external Leary–Minasyan counterexample, and then replaces the proof with new arguments. The main input is the canonical decomposition X ≅ M × R^n × Y supplied by [CM13, Theorem M] and [CM09a, Theorems 1.6/Addendum 1.8]; these are prior published theorems whose assumptions do not include the target statement about commensurated free abelian subgroups, so citing them is independent support rather than circularity. Lemma 6 derives the existence of the commensurated Z^n from the product structure and prior lattice lemmas, and the converse part uses Proposition 4 together with the observation that abelian groups cannot act minimally on non-Euclidean irreducible CAT(0) spaces. No fitted parameter is renamed as a prediction, no definition is stated in terms of the quantity to be derived, and the paper explicitly corrects rather than assumes the false earlier claim. One substantive correctness concern, not a circularity, is that the converse of Theorem 2(i) applies Proposition 4 to H based only on commensuration by the lattice Γ, whereas the proposition requires commensuration by the ambient group; but that is a possible proof gap, not an equation-by-construction equivalence, and it does not make the derivation circular.
Assumptions & free parameters
assumptions (5)
- domain assumption Canonical decomposition of the full isometry group of a proper CAT(0) space with a minimal cocompact closed subgroup: Is(X) = S × (R^n ⋊ O(n)) × D, with S a semisimple Lie group with trivial center and no compact factors, and D totally disconnected.
- standard math Borel density theorem: a lattice in a semisimple Lie group is Zariski dense, and projections of a commensurated abelian subgroup to the semisimple factor are finite.
- domain assumption Balser-Lytchak [BL05, Proposition 1.4]: a group with a fixed closed subset of the visual boundary of circumradius at most π/2 has a fixed point at infinity.
- domain assumption Main result of Caprace, Kropholler, Reid and Wesolek [CK RW]: in a finitely generated group, the profinite closure of a commensurated subgroup contains a finite index subgroup normal in the ambient group.
- standard math Bieberbach theorem and Serre's covolume formula for groups acting on trees.
Cite this review
Pith. "Pith review of Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups"." pith.science (2026). https://pith.science/paper/XU64IGZO
@misc{pith2026190810216,
author = {Pith},
title = {Pith review of: Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups"},
year = {2026},
howpublished = {\url{https://pith.science/paper/XU64IGZO}},
note = {Machine review of arXiv:1908.10216}
}
read the original abstract
We amend the statement of point~(i) in Theorem~1.3 in arxiv:0901.1022 and supply the additional arguments and minor changes for the results that depend on it. We also seize the occasion and generalize to non-finitely generated lattices.
Reference graph
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