Graphical Discreteness, Coxeter Doublings and Generalized Polygons
Year: 2026
Type: preprint
Abstract: Graphical discreteness is not a quasi-isometry invariant in general [16, Corollary 1.3]. For every finite thick generalized m-gon Γ with m ≥ 3, however, it is constant on the class of finitely generated groups quasi-isometric to the right-angled Coxeter group W_Γ. Such a group Λ is graphically discrete if and only if Γ is nonflexible, where nonflexible means that no nontrivial graph automorphism fixes a closed star pointwise [20, p. 1]. The proof combines Fuchsian-building quasi-isometric rigidity with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. The standard Coxeter Cayley-graph criterion is also strengthened from discreteness to compact-by-discreteness. Further results derive obstructions from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and give applications to projective planes and to right-angled Coxeter groups with Menger curve boundary. The arguments do not determine whether any nonflexible finite thick generalized polygon exists.
Source: Preprints.org
Author: Xing-Yu Hu
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204265705