2025/10/03 by Christopher H. Cashen, Cashen, Christopher H., Pallavi Dani +5 · 1 citation
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Artin group #Coxeter complex #Coxeter element #Coxeter graph #Coxeter group #Dimension (graph theory) #Embedding #Geometric and Algebraic Topology #Hyperbolic geometry
paper · pdf · doi:10.1093/imrn/rnag144
openalex created_date 2025/10/09 · openalex publication_date 2026/07/01 · openalex updated_date 2026/08/01
Abstract We introduce a graph-theoretic condition, called (n,m)-branching, that ensures a combinatorial round tree with controlled branching parameters can be quasi-isometrically embedded in the Davis complex of the right-angled Coxeter group defined by the graph. This construction yields a lower bound on the conformal dimension of the boundary of such a hyperbolic group. We exhibit numerous families of graphs with this property, including many 1-dimensional spherical buildings. We prove an embedding result, showing that under mild hypotheses a flag-no-square graph embeds as an induced subgraph in a flag-no-square triangulation of a closed surface. We use this to embed our branching graphs into graphs presenting hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary. We conclude there are examples of such groups with conformal dimension tending to infinity, and hence, there are infinitely many quasi-isometry classes within this family. We use conformal dimension to show that the recent work of Lafont–Minemyer–Sorcar–Stover–Wells can be upgraded to conclude that for every n ≥ 2 there exist infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups that virtually algebraically fiber and have virtual cohomological dimension n.