Graphical Discreteness, Coxeter Doublings and Generalized Polygons
Abstract
Graphical discreteness is not preserved under quasi-isometry in general. We prove that, for every finite thick generalized m-gon Γ with m≥ 3, graphical discreteness is nevertheless constant on the class of finitely generated groups quasi-isometric to the right-angled Coxeter group WΓ: such a group Λ is graphically discrete if and only if Γ is nonflexible, meaning that no nontrivial graph automorphism of Γ fixes a closed star pointwise. The proof combines quasi-isometric rigidity of Fuchsian buildings with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. For finite-rank Coxeter systems, we also show that the automorphism group of the standard Cayley graph is compact-by-discrete if and only if it is discrete, equivalently, if and only if the defining diagram is nonflexible. We further obtain obstructions to graphical discreteness from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and derive applications to finite projective planes and to right-angled Coxeter groups with Menger curve boundary. Our methods do not settle whether nonflexible finite thick generalized polygons exist.
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BibTeX
@misc{hu2026graphical,
title = {{Graphical Discreteness, Coxeter Doublings and Generalized Polygons}},
author = {Xing-Yu Hu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.28550},
eprint = {2609.28550},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28550},
}
RIS
TY - UNPB
TI - Graphical Discreteness, Coxeter Doublings and Generalized Polygons
AU - Xing-Yu Hu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2609.28550
UR - https://arxiv.org/abs/2609.28550
AB - Graphical discreteness is not preserved under quasi-isometry in general. We prove that, for every finite thick generalized m-gon Γ with m≥ 3, graphical discreteness is nevertheless 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, meaning that no nontrivial graph automorphism of Γ fixes a closed star pointwise. The proof combines quasi-isometric rigidity of Fuchsian buildings with a metric-strata argument that recovers the standard Coxeter Cayley graph from the building metric. For finite-rank Coxeter systems, we also show that the automorphism group of the standard Cayley graph is compact-by-discrete if and only if it is discrete, equivalently, if and only if the defining diagram is nonflexible. We further obtain obstructions to graphical discreteness from one-vertex doublings, classify compact-by-discreteness for standard chamber graphs of graph products of finite groups, and derive applications to finite projective planes and to right-angled Coxeter groups with Menger curve boundary. Our methods do not settle whether nonflexible finite thick generalized polygons exist.
ER -