{"id":"c44f91e7-fcc1-4cbf-a9c6-14ee6f2039d3","arxiv_id":"1908.09046","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A subgroup of a right-angled Coxeter group is quasiconvex exactly when its constructed completion complex is finite, and this yields new algorithmic tests for subgroup properties and finite-index embeddability.","lead":"Subgroups of right-angled Coxeter groups can be converted into labeled cube complexes called completions, and the finiteness of these complexes controls whether a subgroup is quasiconvex. The same machinery decides algorithmic questions, such as whether one group embeds inside another with finite index.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 12.1's word-length bound is the load-bearing unverified step; the proof of Theorem 8.4 itself checks out.","rationale":"The reader's weakest assumption is Proposition 12.1, and that is also the single most load-bearing unverified step in the paper's algorithmic claims. The finite-index embeddability algorithm in Theorem 12.8 depends on enumerating exactly the trimmed reflection sets of bounded length; if Proposition 12.1 failed, the algorithm could omit a valid finite-index subgroup and return an incorrect negative answer. The proof of Proposition 12.1 is long and subtle, and no formal or machine verification is provided, so CONDITIONAL remains the appropriate verdict. I did audit the central quasiconvexity theorem, Theorem 8.4. The distance identity in Lemma 8.1, the finite-completion-to-quasiconvexity direction in Lemma 8.2, and the quasiconvexity-to-finite-completion direction in Lemma 8.3 all appear internally consistent. Proposition 3.5 correctly justifies that a finite standard completion is reached in finite time. The omitted proof of Proposition 5.2 is a completeness concern but is not needed for Theorem 8.4. Thus the verdict should stay exactly where the reader placed it: CONDITIONAL, pending independent verification of Proposition 12.1.","tokens_in":47148,"tokens_out":36398,"duration_ms":406123,"concrete_test":"Implement the Section 10 completion and Theorem 6.6 index check. For every triangle-free non-almost-star graph Γ with |V(Γ)| ≤ 6 and every N ≤ 4, enumerate trimmed reduced reflection sets R of size N with word lengths up to an increasing bound L. If any set generates a finite-index subgroup but has max |w_i| > M(V(Γ), N), the bound asserted in Proposition 12.1 fails and the enumeration in Theorem 12.8 is incomplete. If no violation appears, run the same check on random larger graphs as a sanity check.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 12.8 works by enumerating all M-admissible trimmed reflection sets, so its completeness is exactly Proposition 12.1: every trimmed reflection set R of size |V(Γ')| that generates a finite-index subgroup of WΓ must have |w_i| ≤ M, with M depending only on |V(Γ)| and |R|. That proposition is proved by a long chain (Lemmas 12.2–12.5) involving folded trees, the CAT(0) completion Ω_FT, uniqueness of hyperplane intersections, and a maximal-prefix argument. I do not see a concrete error, but the argument is hand-audited only, and the bound is what makes the enumeration finite; a counterexample or a gap in Lemma 12.4 would make the algorithm silently incomplete. This is distinct from the central quasiconvexity theorem: Lemmas 8.1–8.3 and Proposition 3.5 supporting Theorem 8.4 appear internally consistent. The omitted proof of Proposition 5.2 is also not used in Theorem 8.4. The conditional verdict is therefore driven by Proposition 12.1 and not by the headline characterization.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a Stallings-style completion theory for subgroups of right-angled Coxeter groups (RACGs). For a finitely generated subgroup G < W_Γ, the authors construct a Γ-labeled cube complex, called a completion, via fold, cube-attachment, and cube-identification operations. They prove that properties of G are reflected in completions: quasiconvexity is equivalent to finiteness of a standard completion (Theorem A(1) / Theorem 8.4), finite index is characterized by finite full-valence resolved completions (Theorem 6.6), torsion is detected by loops with reduced labels in finite special subgroups (Proposition 4.6), and normality has a core-graph characterization (Theorem 5.3). The paper then applies this machinery to show that finitely generated reflection subgroups are quasiconvex (Theorem 10.5), that one-ended Coxeter subgroups of 2-dimensional RACGs are reflection subgroups and hence quasiconvex (Theorem 11.4 and Corollary 11.5), and to give an algorithm deciding finite-index embeddability between certain RACGs (Theorem 12.8). It also provides algorithms for several properties of quasiconvex subgroups (Theorem E) and gives new proofs of residual finiteness and of Haglund's separability theorem for quasiconvex subgroups.","tokens_in":47401,"tokens_out":5519,"duration_ms":56713,"significance":"If the results are correct, this is a substantial contribution to the geometric and algorithmic study of subgroups of RACGs. The completion construction is a genuinely new tool in this setting, and the characterization of quasiconvexity by finiteness of a completion is both conceptually clean and algorithmically useful. The paper contains many explicit, detailed arguments, and the main theorems are not obtained by circular reasoning: the completion is defined independently, and the axioms ledger is empty. The algorithmic applications, especially Theorems D and E, are strong and well-motivated. However, the paper's central algorithmic theorem depends on a long and only hand-audited combinatorial bound, and several statements rely on omitted proof details. These points need to be addressed before the paper can be accepted in its present form.","major_comments":[{"comment":"The completeness of the algorithm in Theorem 12.8 rests entirely on Proposition 12.1, because the algorithm enumerates M-admissible trimmed reflection sets and would silently miss a finite-index subgroup if the uniform length bound failed. The proof of Proposition 12.1 is a long chain of auxiliary lemmas, and I could not fully verify the crucial uniqueness claim in Lemma 12.4, namely that ``e_1 is the only edge of ``f(FT) dual to H_1``. In particular, the step where the maximal-prefix choice of the expressions w_i is used to rule out all other edges dual to H_1 deserves a more explicit and self-contained justification. Since a gap here would invalidate the main algorithmic theorem, I request that the proof of Proposition 12.1 be expanded, with special attention to Lemma 12.4 and to the maximal-prefix argument.","section":"Section 12, Proposition 12.1 and Lemmas 12.2–12.5"},{"comment":"Proposition 5.2 is stated with the proof omitted, with the explanation that it follows closely from [KM02, Theorem 5.2]. This is not satisfactory, because in the present setting completions are not unique and need not even have the same homotopy type (Example 3.7), so uniqueness of the core graph is a nontrivial claim. The proposition is used in the proof of Theorem 5.3, which is one of the advertised characterizations in Theorem A(2). Please provide a complete proof, or at minimum a precise reduction to [KM02, Theorem 5.2] that accounts for the distinction between completions and Stallings graphs.","section":"Section 5, Proposition 5.2"},{"comment":"In the fold operation case, the proof says that path types 3 and 4 are handled similarly and omits them. These cases are not merely cosmetic: they cover loops based at the basepoint B that traverse the folded edge near the beginning or end of the loop, and they are needed for the iteration in Lemma 3.10, which in turn underpins Theorem 3.11. Please supply the missing arguments for types 3 and 4.","section":"Section 3, Lemma 3.9"},{"comment":"The introduction states Theorem D as providing an algorithm that, given a one-ended 2-dimensional RACG W_Γ and any RACG W_Γ', decides whether W_Γ' embeds as a finite-index subgroup of W_Γ. However, the theorem actually proved in Section 12 assumes that Γ' has no isolated vertex. If the isolated-vertex case is genuinely excluded, the introductory and abstract statements need to be revised; if it is covered by a separate argument, that argument must be included. This mismatch concerns one of the paper's central advertised claims.","section":"Introduction, Theorem D; Section 12, Theorem 12.8"}],"minor_comments":[{"comment":"The statement contains a typo: the right-hand side reads ``C_2(Ω_1, B_2)``, but it should presumably be ``C_2(Ω_2, B_2)``.","section":"Section 5, Proposition 5.2"},{"comment":"The two completions are described as a torus and a Klein bottle obtained by attaching a square to the rose graph, but the attaching maps are not written out. Since the labels and the attaching words determine whether the complexes are Γ-labeled completions, the example would be much easier to check if the boundary words of the attached squares were specified explicitly.","section":"Section 3, Example 3.7"},{"comment":"Several figures omit edge labels or use very small labels, which makes the examples difficult to verify independently. Please add labels or explain the omitted labels in the captions.","section":"Figures 2 and 7"},{"comment":"In case (iv) of the proof, the reduction to the non-almost-star case via the kernel K' is clear in principle, but the sentence ``The theorem now follows`` hides the induction on the number of vertices. Since the algorithm is central, a short explicit statement of the induction and of why termination is guaranteed would improve readability.","section":"Section 12, Theorem 12.8"},{"comment":"The notation ``k^{(n mod 2)}`` is used without a prior definition. For clarity, state explicitly that this means the word k if n is odd and the empty word if n is even.","section":"Section 13, Lemma 13.1"}],"recommendation":"major_revision","confidential_remarks":"This is a strong paper with a novel and promising framework, but the current version is not yet suitable for publication. The main concerns are the unverified load-bearing argument in Proposition 12.1, the omitted proof of Proposition 5.2, and the incomplete cases in Lemma 3.9. I would be willing to look at a revised version that expands these points. I also recommend that the authors check the consistency between Theorem D in the introduction and Theorem 12.8, since the advertised generality is not what is proved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things. First, the completion construction is the real thing, not a repackaging. Second, the headline quasiconvexity theorem is in good shape; the place to worry is Section 12, where the completeness of the embeddability algorithm depends on a word-length bound that is only hand-audited.\n\nWhat is actually new: completions are folded, cube-full cube complexes built from a rose by explicit operations. They have a reduced-word loop property that Stallings-like graphs do not, and they work without any hyperbolicity assumption. The central result, Theorem 8.4, is that a subgroup of a RACG is quasiconvex iff it is finitely generated and some (equivalently every) standard completion is finite. I checked the distance comparison in Lemma 8.1 and the geodesic-ray argument in Lemma 8.3; both are internally consistent, and the proof of the characterization holds up. Theorems B and C are substantial applications: reflection subgroups are quasiconvex, and one-ended Coxeter subgroups of 2-dimensional RACGs are reflection subgroups. The new proof of Haglund's separability theorem via full-valence extensions is also clean and worth having.\n\nSoft spots, in order of importance. Proposition 12.1 is the load-bearing step for the finite-index embeddability algorithm in Theorem 12.8. The algorithm enumerates exactly the M-admissible trimmed reflection sets, and if the bound failed, the enumeration would silently miss the subgroup. The proof is a long chain through folded trees, the CAT(0) completion, hyperplane uniqueness, and a maximal-prefix argument. I do not see a concrete error, and the lemmas are plausible, but this is precisely the kind of combinatorial-geometric bound that can hide a gap. It needs either a very careful referee or a more modular verification. Two proofs are explicitly omitted: Proposition 5.2 (core graph uniqueness) and parts of Lemma 3.9 (fold path types 3 and 4). Both are needed for the complete story but not for Theorem 8.4; I believe they can be filled without changing the framework. Minor: in Theorem 12.8 case (iv), the computation of generators for the kernel K' is not spelled out, but it looks doable.\n\nWho this is for: anyone working on subgroup structure of RACGs, algorithmic geometric group theory, or Stallings-style methods in cubical groups. The paper deserves a serious referee; the core is solid and new, and the omitted proofs are fillable. Send it out, and ask the authors to write up the omitted arguments and to expand or verify Proposition 12.1.","headline":"A genuinely useful Stallings-style completion framework for RACGs; the quasiconvexity characterization is solid, but the finite-index embeddability algorithm leans on a long hand-audited word-length bound.","tokens_in":47895,"tokens_out":2155,"would_cite":true,"duration_ms":24668,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","57M07","20F55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that a subgroup of a right-angled Coxeter group is quasiconvex exactly when its standard completion—an edge-labeled cube complex built from the subgroup's generators—is finite, and uses this equivalence to detect…","keywords":["right-angled Coxeter groups","quasiconvex subgroups","Stallings-like techniques","cube complexes","reflection subgroups","finite-index subgroups","algorithmic group theory","separability"],"falsifier":"Find a triangle-free, non-almost-star graph Γ and a finite set of reflections R generating a finite-index subgroup of WΓ such that every trimmed generating set contains a word longer than the constant M(|V(Γ)|,|R|) supplied by Proposition 12.1; equivalently, exhibit a pair (Γ,Γ′) for which the algorithm of Theorem 12.8 says 'no' although WΓ′ embeds as a finite-index subgroup of WΓ.","tokens_in":46984,"feed_emoji":"🧊","tokens_out":6854,"duration_ms":64823,"temperature":0.7,"pith_summary":"The paper introduces 'completions' for subgroups of right-angled Coxeter groups (RACGs): cube complexes built from a finite generating set by Stallings-style folds, cube attachments, and cube identifications. Its central theorem states that a subgroup is quasiconvex if and only if it is finitely generated and every standard completion is finite. The same complexes encode finite index, torsion, and normality, and they turn those properties into algorithmically checkable conditions. The authors use this to prove that reflection subgroups and one-ended Coxeter subgroups of 2-dimensional RACGs are quasiconvex, to give an algorithm deciding finite-index embeddability between RACGs, and to re-prove Haglund's separability theorem.","feed_headline":"Quasiconvex subgroups are exactly the ones with finite completions","feed_subtitle":"Cube-complex completions make index, normality, and torsion of subgroups algorithmically checkable.","key_machinery":"The completion is the central object. Starting from a 'rose' graph whose petals are labeled by the chosen generator words, one repeatedly folds pairs of edges with the same label, attaches cubes whenever a tuple of incident edges has labels forming a clique in the defining graph, and identifies cubes with identical boundaries; the process stops when the complex is folded and cube-full. Being cube-full guarantees that no commuting relations are missing, so reduced words behave like geodesics in the complex; this is what lets finiteness of the completion control quasiconvexity and lets non-positively curved completions support the geometric arguments.","core_discovery":"For any finitely generated subgroup G of a RACG WΓ, the paper constructs a completion Ω: a folded, cube-full, edge-labeled cube complex whose loops based at the basepoint carry exactly the reduced words representing elements of G, and in which every reduced word for a group element labels a loop. Theorem 8.4 states that G is quasiconvex in WΓ if and only if G is finitely generated and some (equivalently every) standard completion is finite. A completion is finite exactly when it can be built in finite time, so quasiconvexity becomes decidable. The paper further shows that a completion determines whether G is finite-index (the complex is finite and every vertex is incident to an edge of every label), torsion-free (no loop reduces into a finite special subgroup), and normal (conditions on the core graph, with a computable reformulation).","pith_inferences":["A natural next step, not taken in the paper, is to implement the standard completion algorithm on small defining graphs and compare the finite-completion criterion with known commensurability classifications; this would give empirical data on how sharp the length bound in Proposition 12.1 is.","Because the completion construction does not assume hyperbolicity, the same cube-full finiteness criterion may be adaptable to other graph products of groups whose defining graphs carry a similar commutativity structure, though the paper does not claim this.","The algorithm in Theorem 12.8 effectively reduces finite-index embeddability to a bounded search over trimmed reflection sets; if the bound in Proposition 12.1 can be made explicit and small, the theorem becomes a practical computational tool for commensurability questions."],"forward_implications":["Quasiconvexity of a subgroup of a RACG is algorithmically detectable: build a standard completion and check whether it is finite.","For a quasiconvex subgroup given by finitely many words, there are algorithms to test torsion-freeness, compute its index, test whether a power of a given element lies in the subgroup, and test normality.","Every finitely generated reflection subgroup of a RACG is quasiconvex, because it admits a finite completion.","Every one-ended Coxeter subgroup of a 2-dimensional RACG is a reflection subgroup and hence quasiconvex.","There is an explicit algorithm that decides, given a 2-dimensional RACG WΓ and any RACG WΓ′, whether WΓ′ is isomorphic to a finite-index subgroup of WΓ, and outputs the embedding words when it is."],"supporting_citations":[{"why":"Supplies the combinatorial Stallings machinery and the core-graph uniqueness argument that completions generalize from free groups to RACGs.","marker":"[KM02]"},{"why":"Introduces the Stallings folds and canonical labeled graphs whose cube-complex analogue is developed here.","marker":"[Sta83]"},{"why":"Used to conclude that a quasiconvex subgroup of a finitely generated group is finitely generated, a step in Theorem 8.4.","marker":"[BH99]"},{"why":"A prior Stallings-style characterization of quasiconvex subgroups in cubulated hyperbolic groups, contrasted with the non-hyperbolic RACG setting of this paper.","marker":"[BL18]"},{"why":"Proves reflection subgroups of Coxeter groups are Coxeter and describes standard reflection generating sets, used in Theorems C and 12.8.","marker":"[Dye90]"},{"why":"Independent proof that reflection subgroups are Coxeter, supporting the same reflection-subgroup results.","marker":"[Deo89]"},{"why":"Established that quasiconvex subgroups of RACGs are separable and virtual retracts, a result reproved here via completions.","marker":"[Hag08]"},{"why":"Shows a right-angled Coxeter group determines its defining graph, used by the algorithm to recognize when a generated subgroup is isomorphic to the target RACG.","marker":"[Rad03]"}],"fun_headline_variants":["Completions decode Coxeter subgroups: quasiconvex, finite-index, normal","Finite completion equals quasiconvex in right-angled Coxeter groups","Quasiconvexity check: build the completion, see if finite","Stallings-style cubes reveal Coxeter subgroup properties","Coxeter subgroups: finite completions mark quasiconvexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The finite-index embeddability algorithm rests on the bound in Proposition 12.1: for a triangle-free defining graph that is not almost star, every trimmed reflection generating set of a finite-index subgroup has word lengths bounded by a constant depending only on the graph and the number of reflections, so a subgroup needing a longer generator would escape the algorithm's enumeration.","fun_headline_variants_meta":{"raw":{"variants":["Completions decode Coxeter subgroups: quasiconvex, finite-index, normal","Finite completion equals quasiconvex in right-angled Coxeter groups","Quasiconvexity check: build the completion, see if finite","Stallings-style cubes reveal Coxeter subgroup properties","Coxeter subgroups: finite completions mark quasiconvexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000274,"raw_usage":{"total_tokens":1584,"prompt_tokens":834,"completion_tokens":750,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":450,"completion_tokens_details":{"reasoning_tokens":657}},"tokens_in":450,"tokens_out":750,"duration_ms":8021,"temperature":1.0,"reasoning_tokens":657,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:25:15.045406+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a triangle-free, non-almost-star graph Γ and a finite set of reflections R generating a finite-index subgroup of WΓ such that every trimmed generating set contains a word longer than the constant M(|V(Γ)|,|R|) supplied by Proposition 12.1; equivalently, exhibit a pair (Γ,Γ′) for which the algorithm of Theorem 12.8 says 'no' although WΓ′ embeds as a finite-index subgroup of WΓ.","supporting_citations":[],"review_version":1}