{"id":"17e38beb-e2f8-4e80-89fa-6c3b51ef1f06","arxiv_id":"2510.03430","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"(n,m)-branching graphs give right-angled Coxeter groups with boundary conformal dimension at least 1 + log n/log(3m-7), yielding infinitely many quasi-isometry classes with Pontryagin sphere boundary and with virtual algebraic fibering in every virtual cohomological dimension >= 2.","lead":"This paper introduces a graph condition called (n,m)-branching that forces large conformal dimension of the boundary of a right-angled Coxeter group, and uses it to build infinitely many quasi-isometry types of groups with Pontryagin sphere boundary and of groups that virtually algebraically fiber. A general reader might care because it tells apart many hyperbolic groups that previous tools could not distinguish, and it upgrades known fibering results to quasi-isometry classi","discovery_kind":"new_method","skeptic_critique":null,"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["20F65","20F55","57M07","30L10","51E24"],"pacs":[],"model":"deepseek-v4-flash","headline":"Branching graphs split Coxeter groups into infinitely many types","keywords":["right-angled Coxeter groups","hyperbolic groups","conformal dimension","quasi-isometry classification","Pontryagin sphere","algebraic fibering","virtual cohomological dimension","round trees"],"falsifier":"A concrete test: for a small (n,m)-branching graph such as the Heawood graph, explicitly construct the first two stages of the round tree described in the paper and verify that each link of the new subcomplex is an induced subgraph of the defining graph. If any link contains an induced 4-cycle or an unwanted edge that breaks convexity, the lower bound would fail for that graph. Likewise, a computer search over finite graphs of girth 5 could look for one that cannot be embedded as an induced subgraph of any flag-no-square triangulation of a closed orientable surface; the existence of such a gra","tokens_in":22429,"feed_emoji":"🌐","tokens_out":11310,"duration_ms":122393,"temperature":0.7,"pith_summary":"This paper introduces a local graph-theoretic condition, (n,m)-branching, that controls how the Gromov boundary of a hyperbolic right-angled Coxeter group behaves: any graph with this condition defines a group whose boundary has conformal dimension at least 1 + log n / log(3m−7). The proof works by embedding a combinatorial round tree into the group's cube complex with vertical branching n and horizontal branching at most 3m−7, then applying known bounds that turn round-tree parameters into conformal dimension estimates. From this, the authors obtain two families of results: hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary whose conformal dimensions tend to infinity, which means infinitely many quasi-isometry classes; and, for every dimension n ≥ 2, infinitely many quasi-isometry classes of virtually algebraically fibered hyperbolic right-angled Coxeter groups of virtual cohomological dimension n. The significance is that conformal dimension—a quasisymmetry invariant of the boundary—is used to separate quasi-isometry classes within families that were previously not known to contain infinitely many.","feed_headline":"Branching graphs split Coxeter groups into infinitely many types","feed_subtitle":"A graph condition yields infinitely many quasi-isometry classes, including groups with Pontryagin sphere boundaries.","key_machinery":"The key object is the combinatorial round tree, a 2-complex built from a nested sequence of disks that branch n-fold in the vertical direction while each disk meets at most H new disks in the next level; here H = 3m−7. The (n,m)-branching condition on the defining graph supplies, at every vertex of the outer edge path of the current disk, n distinct cycles of squares of length at most m that share exactly the current segment and whose union is an induced subgraph of the defining graph. This is precisely what makes the growing subcomplex locally convex in the cube complex, so it is a quasiconvex round tree. The count 3m−7 is the worst-case number of new squares that a single square can meet w","core_discovery":"The central claim is that the (n,m)-branching condition, which requires every vertex to have degree at least n+1 and every induced edge or two-edge segment to be extendable to n cycles of length between 5 and m whose pairwise intersection is exactly the segment and whose union is induced, forces the conformal dimension of the boundary of the hyperbolic right-angled Coxeter group WΓ to be at least 1 + log n / log(3m−7). The proof constructs a combinatorial round tree as a convex subcomplex of the cube complex associated to WΓ, with vertical branching n and horizontal branching at most 3m−7; the branching condition is exactly what is needed to keep the subcomplex locally convex while new squar","pith_inferences":["The horizontal branching count 3m−7 comes from a worst-case analysis of how a square meets the outer edge path; a sharper count for specific families—for instance, when all cycles are hexagons—might lower H and thus raise the conformal dimension lower bound for the same graphs.","The surface-embedding lemma is only sketched in the paper: the proof shows how to get the graph as a subcomplex of a triangulation but does not explicitly justify that the triangulation can be made flag-no-square while preserving the graph as an induced subcomplex. If that gap cannot be filled, the Pontryagin-sphere family would still be plausible but would need a different construction.","The fibering upgrade works by making the group a lattice in a thick building with a prescribed underlying Coxeter group; this suggests a general recipe—any hyperbolic right-angled group family that can be thickened to large multiplicity with fixed virtual cohomological dimension will automatically contain infinitely many quasi-isometry classes of virtually fibered groups.","Because the branching condition is purely local and the round tree is quasiconvex, the conformal dimension lower bound applies not only to the whole group but to any supergroup obtained by a quasi-isometric embedding that respects the tree; this could transfer the bounds to other classes of groups containing such Coxeter subgroups."],"forward_implications":["For m fixed and n growing, the sequence of (n,m)-branching graphs yields hyperbolic right-angled Coxeter groups whose boundary conformal dimension grows without bound, hence infinitely many quasi-isometry classes.","Embedding these graphs as induced subgraphs of flag-no-square surface triangulations gives infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary.","For every virtual cohomological dimension n ≥ 2, there are infinitely many quasi-isometry classes of virtually algebraically fibered hyperbolic right-angled Coxeter groups.","The (n,m)-branching condition also forces geometric consequences: such graphs have girth at least 5, are inseparable for n ≥ 2, and are nonplanar for n ≥ 3, so the boundary of the group is a planar Sierpinski carpet or a Menger curve depending on the degree of branching and planarity.","For each fixed n, the lower bound 1 + log n / log(3m−7) weakens as m grows, so the families with the strongest control (smallest m, such as the hexagon-based examples with m = 6) give the best bounds."],"fun_headline_variants":["Branching graphs bound conformal dimension, split Coxeter groups","Infinite quasi-isometry classes from branching graph embeddings","Pontryagin sphere boundary: branching yields infinite Coxeter types","Branching condition drives conformal dimension to infinity"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The proof depends on the unproved claim that any girth-at-least-5 graph embeds as an induced subgraph of a flag-no-square triangulation of a closed orientable surface, together with the local convexity checks in the round-tree construction.","fun_headline_variants_meta":{"raw":{"variants":["Branching graphs bound conformal dimension, split Coxeter groups","Infinite quasi-isometry classes from branching graph embeddings","Pontryagin sphere boundary: branching yields infinite Coxeter types","Branching condition drives conformal dimension to infinity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000581,"raw_usage":{"total_tokens":2585,"prompt_tokens":770,"completion_tokens":1815,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1761}},"tokens_in":514,"tokens_out":1815,"duration_ms":70968,"temperature":1.0,"reasoning_tokens":1761,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T11:42:21.505518+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete test: for a small (n,m)-branching graph such as the Heawood graph, explicitly construct the first two stages of the round tree described in the paper and verify that each link of the new subcomplex is an induced subgraph of the defining graph. If any link contains an induced 4-cycle or an unwanted edge that breaks convexity, the lower bound would fail for that graph. Likewise, a computer search over finite graphs of girth 5 could look for one that cannot be embedded as an induced subgraph of any flag-no-square triangulation of a closed orientable surface; the existence of such a gra","supporting_citations":[],"review_version":1}