发表机构
School of Mathematics and Statistics, Hanjiang Normal University（汉江师范大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明在拟等距于右角Coxeter群的有限生成群类中，图形离散性等价于广义多边形的非柔性，并结合建筑刚性与度量分层给出分类及若干应用。
AI 中文摘要
图形离散性在拟等距下一般不保持。我们证明，对于每个有限厚广义 $m$-边形 $\Gamma$（$m\geq 3$），图形离散性在拟等距于右角 Coxeter 群 $W_\Gamma$ 的有限生成群类上仍然是常数：这样的群 $\Lambda$ 是图形离散的当且仅当 $\Gamma$ 是非柔性的，即 $\Gamma$ 的没有非平凡图自同构逐点固定一个闭星。证明结合了 Fuchsian 建筑的拟等距刚性以及一个度量分层论证，该论证从建筑度量中恢复标准 Coxeter Cayley 图。对于有限秩 Coxeter 系统，我们还证明了标准 Cayley 图的自同构群是紧致-离散的当且仅当它是离散的，等价地，当且仅当定义图是非柔性的。我们进一步从单顶点加倍获得图形离散性的障碍，对有限群图乘积的标准室图分类紧致-离散性，并推导出对有限射影平面和具有 Menger 曲线边界的右角 Coxeter 群的应用。我们的方法未解决非柔性有限厚广义多边形是否存在。
英文摘要
Graphical discreteness is not preserved under quasi-isometry in general. We prove that, for every finite thick generalized $m$-gon $Γ$ with $m\geq 3$, 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.
Comments21 pages
