Hyperbolic models and stability recognition for Coxeter groups
Abstract
Given a finite rank Coxeter system, we construct a hyperbolic space by coning off the wide parabolic subcomplexes of its Davis complex. The resulting space is ``stability recognizing'', in the sense that stable subspaces of the Davis complex are exactly the quasigeodesically connected subspaces whose images in the coned-off space are quasiisometrically embedded. Equivalently, the coned-off space is ``Morse recognizing'': a quasigeodesic in the Davis complex is Morse if and only if it is quasigeodesic in the coned-off space. Similar results are true for the mapping class group of a hyperbolic surface acting on its curve graph. We strengthen this analogy with additional results. First, the action of the Coxeter group on the coned-off space is acylindrical and gives the largest acylindrically hyperbolic structure for the Coxeter group. Second, the stable subgroups of a Coxeter group are characterized as the finitely generated subgroups that are undistorted and purely loxodromic with respect to the action on the coned-off space. The coned-off space, being constructed directly from the Coxeter structure, facilitates concrete characterizations of Morseness and stability for two important Coxeter-native families of objects: combinatorial geodesics and special subgroups.
Related
- Dimension bounds for finite-volume hyperbolic Coxeter polytopes
- Compact Hyperbolic Coxeter Six-dimensional Polytopes With Ten Facets
- On algebraic dynamics of birational automorphisms of Calabi-Yau manifolds and the universal Coxeter groups
- Property R∞ for Artin and Coxeter groups whose outer automorphisms are length-preserving
- Bounded Bruhat intervals and affine Coxeter groups
BibTeX
@misc{cashen2026hyperbolic,
title = {{Hyperbolic models and stability recognition for Coxeter groups}},
author = {Christopher H. Cashen and Michelle Chu and Jing Tao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.17036},
eprint = {2608.17036},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.17036},
}
RIS
TY - UNPB
TI - Hyperbolic models and stability recognition for Coxeter groups
AU - Christopher H. Cashen
AU - Michelle Chu
AU - Jing Tao
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2608.17036
UR - https://arxiv.org/abs/2608.17036
AB - Given a finite rank Coxeter system, we construct a hyperbolic space by coning off the wide parabolic subcomplexes of its Davis complex. The resulting space is ``stability recognizing'', in the sense that stable subspaces of the Davis complex are exactly the quasigeodesically connected subspaces whose images in the coned-off space are quasiisometrically embedded. Equivalently, the coned-off space is ``Morse recognizing'': a quasigeodesic in the Davis complex is Morse if and only if it is quasigeodesic in the coned-off space. Similar results are true for the mapping class group of a hyperbolic surface acting on its curve graph. We strengthen this analogy with additional results. First, the action of the Coxeter group on the coned-off space is acylindrical and gives the largest acylindrically hyperbolic structure for the Coxeter group. Second, the stable subgroups of a Coxeter group are characterized as the finitely generated subgroups that are undistorted and purely loxodromic with respect to the action on the coned-off space. The coned-off space, being constructed directly from the Coxeter structure, facilitates concrete characterizations of Morseness and stability for two important Coxeter-native families of objects: combinatorial geodesics and special subgroups.
ER -