{"id":"d6e8e9e7-033b-4b72-b707-f92ec87221cf","arxiv_id":"2607.08356","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The maximal number of bushy free factors that coarsely embed into a standard HHS is the orthogonality number of its bushy domains; this distinguishes Torelli groups, Johnson kernels and surface braid groups.","lead":"The paper bounds how many bushy free-group factors can coarsely embed into mapping class groups, Torelli groups, Johnson kernels, surface braid groups and Bestvina-Brady groups, via hierarchical hyperbolicity. These bounds distinguish the groups up to quasi-isometry and obstruct coarse embeddings where asymptotic dimension and cohomological dimension fail.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader correctly isolates the modelling hypothesis (standard HHS) as the only non-trivial modelling choice and notes that it holds for every family to which the paper is applied. The technical core—counting points that become close after factoring out quasilines (Prop. 4.2) and then producing a bilipschitz flat by induction plus geometric analysis (Thm. 3.1)—is written in full and re-uses only well-established HHS tools. No further load-bearing weakness appears; the suggested entropy check is merely a routine numerical sanity test of an already-cited combinatorial estimate. Consequently the ACCEPT verdict stands.","tokens_in":36171,"tokens_out":408,"duration_ms":5157,"concrete_test":"Verify that the binary-entropy bound of Lemma 4.5, once inserted into the product estimate at the end of the proof of Proposition 4.2, indeed yields an arbitrary a>1 for sufficiently small ε; if the resulting exponent cannot be made smaller than a, the “most directions” control fails and the induction cannot start.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (Theorem 3.1) and its corollaries rest on the standard-HHS hypotheses (Definition 2.3) and on the counting/induction arguments of Sections 4–5. Those hypotheses are verified for the intended examples (MCGs via Lemma 2.4, RAAGs/Bestvina–Brady via the existing HHS structures of [BHS17b]), the distance-formula and gate-map estimates are standard, and the geometric-analysis steps (full-rank Jacobians upgraded by underspill) follow the same pattern already used for quasiflats in [BHS17b, §13]. No internal gap or unstated assumption that would invalidate the bilipschitz flat or the resulting orthogonality bound was found.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper introduces the maximal number of factors of a product of bushy trees (or free groups) that coarsely or quasi-isometrically embed into a space as a quasi-isometry invariant, and computes it for mapping class groups, Torelli groups, Johnson kernels, surface braid groups and Bestvina–Brady groups. The core technical result (Theorem 3.1) asserts that a coarse embedding of the product of k regular 3-valent trees into a standard hierarchically hyperbolic space X produces a standard k-flat in an asymptotic cone of the factored space of X; consequently k is bounded by the maximal number of pairwise orthogonal bushy domains (Corollary 7 and the refined Corollary 3.6). These bounds are then used to distinguish quasi-isometry classes and to obstruct coarse embeddings that are invisible to asymptotic dimension or virtual cohomological dimension.","tokens_in":36358,"tokens_out":735,"duration_ms":7813,"significance":"The work supplies a new, monotonic quasi-isometry invariant that is strictly finer than quasiflat rank for several important classes of groups. The applications to Torelli groups versus Johnson kernels (Corollary 3) and to surface braid groups of different genera (Theorem 4) are new and cleanly separate groups that share the same asymptotic dimension and virtual cohomological dimension. The technical machinery—counting arguments that control the factored map, followed by an inductive geometric-analysis argument that upgrades full-rank Jacobians to bilipschitz flats—extends the quasiflat techniques of earlier HHS papers and is likely to be reusable for other cubulated or hierarchically hyperbolic groups. The results are parameter-free once the standard-HHS hypotheses are verified, and the verification for the intended examples is supplied.","major_comments":[],"minor_comments":[{"comment":"In the proof of Lemma 4.6 the constant C is required to be “sufficiently large in terms of the HHS parameters only,” but the precise lower bounds needed for the realisation theorem and the partial order on relevant domains are never collected in one place; a short list of the inequalities imposed on C would make the argument easier to check.","section":null},{"comment":"Figure 1 and Figure 2 are helpful, yet the captions do not record the precise Euler-characteristic calculation that yields the floor((3g+p-2)/2) bound; a one-line reference to the argument of Lemma 3.7 would remove any ambiguity.","section":null},{"comment":"The notation for the factored metric and the factored distance-formula sums (ˆd, ˆσ) is introduced in Definition 2.3 and Remark 2.14, but is occasionally reused without the hat when the ambient space is already the factored space; a consistent convention would improve readability.","section":null},{"comment":"In Section 5.1 the appeal to the co-area formula and to Eilenberg–Harrold is correct, yet the precise statement of the co-area formula used (Sim18, Thm. 2.7.3) is not reproduced; a one-sentence reminder of the hypotheses would help readers less familiar with geometric measure theory.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is long and technical, but the central derivation is complete and the applications are of genuine interest to geometric group theorists working with mapping class groups and RAAGs. I see no reason to delay publication."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is that the maximal number of factors of a product of bushy trees that coarsely embed into a standard HHS is controlled by the maximal set of pairwise orthogonal bushy domains (Corollary 7, via the bilipschitz flat in Theorem 3.1). That number is strictly smaller than the quasiflat rank for MCGs, and it cleanly separates Torelli from Johnson kernels (Corollary 3), certain surface braid groups (Theorem 4), and Bestvina-Brady kernels that share finiteness, Dehn function and asdim (Theorem 3.10). It also obstructs some coarse embeddings between MCGs and RAAGs that vcd/asdim do not catch.\n\nWhat is new is the choice of invariant itself (products of bushy trees rather than flats) together with the technical route: a sharp counting argument that the factored map still behaves like a quasi-isometric embedding in most directions (Proposition 4.2, Lemmas 4.5–4.8), followed by an inductive geometric-analysis upgrade from full-rank Jacobians to bilipschitz maps on asymptotic cones (Section 5). The applications then follow by verifying incompatibility of certain domains (one-holed tori for Torelli, etc.) and feeding the resulting orthogonality numbers into the general bound.\n\nThe soft spots are minor and local. The whole machine needs the “standard HHS” package (Definition 2.3: bushiness of non-minimal domains, quasilines at the bottom, bounded geometry). That is verified for the intended examples (Lemma 2.4 for HHGs, existing structures for RAAGs/Bestvina-Brady), but it is a modelling choice; without bushiness the counting lemmas fail. The surface-braid bounds are not optimal, and the handlebody conjecture is left open. None of this touches the central derivation.\n\nThis is for people who work with HHSs, mapping class groups, or coarse embeddings of RAAGs/Coxeter groups. The math is pure, fully written, and rests on standard distance-formula and median-geometry tools; the citation pattern is normal self-citation of background lemmas. I would send it to a serious referee without hesitation.","headline":"Clean new QI invariant from products of bushy trees that separates Torelli/Johnson kernels and certain braid/Bestvina-Brady groups where asdim and vcd fail; proofs look solid.","tokens_in":36924,"tokens_out":547,"would_cite":true,"duration_ms":6468,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","57M07","20F67","57K20"],"pacs":[],"model":"grok-4.5","headline":"The largest product of bushy trees that embeds into an HHS is the max number of pairwise orthogonal bushy domains; this distinguishes mapping class groups, Torelli groups, Johnson kernels and more.","keywords":["hierarchically hyperbolic spaces","coarse embeddings","products of trees","quasi-isometry invariants","mapping class groups","Torelli groups","Johnson kernels","Bestvina-Brady groups"],"falsifier":"Exhibit a coarse embedding of a product of more free groups of rank 2 than the orthogonality number of bushy domains into a standard HHS (for instance into a mapping class group), or show that the factored map fails to be quasi-isometric in most directions for some standard HHS.","tokens_in":37088,"feed_emoji":"🌳","tokens_out":592,"duration_ms":6130,"temperature":0.7,"pith_summary":"This paper proposes a new quasi-isometry invariant: the largest number of factors of a product of bushy trees (or free groups of rank 2) that can coarsely embed into a space. For a standard hierarchically hyperbolic space the number is at most the size of a largest set of pairwise orthogonal bushy domains in its hierarchy. The authors prove this by showing that any coarse embedding of a product of k regular trees produces a bilipschitz k-flat in an asymptotic cone of the factored space (the HHS with its quasilines discarded). They then compute the number for mapping class groups, Torelli groups, Johnson kernels, surface braid groups and certain Bestvina-Brady groups, and use the resulting values to separate those groups up to quasi-isometry and to obstruct coarse embeddings between them. The invariant is monotonic under coarse embeddings and subgroups, unlike the usual quasiflat rank, and already distinguishes pairs that virtual cohomological dimension and asymptotic dimension cannot.","feed_headline":"Tree products measure the orthogonal rank of HHS groups","feed_subtitle":"The count separates Torelli groups from Johnson kernels and blocks many coarse embeddings.","key_machinery":"Theorem 3.1: a coarse embedding of T_3^k into a standard HHS yields, after passage to the factored space and asymptotic cones, a bilipschitz k-flat that is the ultralimit of hierarchy boxes supported on pairwise orthogonal bushy domains.","core_discovery":"Any coarse embedding of a product of k bushy trees into a standard hierarchically hyperbolic space X produces a standard k-flat in an asymptotic cone of the factored space of X; therefore k cannot exceed the maximal number of pairwise orthogonal bushy domains.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Bushy tree products measure orthogonal rank of HHS groups","Max embeddable bushy trees equal orthogonal domains in HHS","Tree products separate Torelli groups from Johnson kernels","Coarse embeddings of tree products bound HHS orthogonal rank","Products of bushy trees yield quasi-isometry invariants for HHS"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The HHS must be standard: every unbounded non-minimal domain is bushy, every non-bushy unbounded domain is a quasiline, and the space has bounded geometry.","fun_headline_variants_meta":{"raw":{"variants":["Bushy tree products measure orthogonal rank of HHS groups","Max embeddable bushy trees equal orthogonal domains in HHS","Tree products separate Torelli groups from Johnson kernels","Coarse embeddings of tree products bound HHS orthogonal rank","Products of bushy trees yield quasi-isometry invariants for HHS"]},"model":"grok-4.5","effort":"low","cost_usd":0.008022,"raw_usage":{"total_tokens":1799,"prompt_tokens":597,"num_sources_used":0,"completion_tokens":87,"cost_in_usd_ticks":80220000,"prompt_tokens_details":{"text_tokens":597,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1115,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":597,"tokens_out":87,"duration_ms":9786,"temperature":1.0,"reasoning_tokens":1115,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T08:47:54.396310+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a coarse embedding of a product of more free groups of rank 2 than the orthogonality number of bushy domains into a standard HHS (for instance into a mapping class group), or show that the factored map fails to be quasi-isometric in most directions for some standard HHS.","supporting_citations":[],"review_version":1}