论文标题
带有凸球的图
Graphs with convex balls
论文作者
论文摘要
在本文中，我们研究了所有球都是凸的图，以及几何作用于它们（我们称为CB图和CB组）的组。 Soltan和Chepoi（1983）以及Farber和Jamison（1987）引入和表征了这些图。 CB图形和CB组概括了收缩期（别名桥接）和弱收缩期图和组，它们在几何组理论中起着重要作用。 我们提出了CB图形的度量和局部到全球表征。也就是说，我们将cb-graphs $ g $描述为图形，其三角形pentagonal综合体$ x（g）$是仅连接的，半径的球最多是$ 3 $。与收缩期和弱收缩图类似，我们证明了CB-Graphs $ G $的拆卸性结果：我们证明他们的正方形$ g^2 $是可拆卸的。这意味着CB图纸的撕裂络合物是可缩度的。最后，我们适应并扩展了Januszkiewicz和Swiatkowski（2006）的收缩期组和Chalopin等人的方法。 （2020年）对于Helly组，表明CB组是双重的。
In this paper, we investigate the graphs in which all balls are convex and the groups acting on them geometrically (which we call CB-graphs and CB-groups). These graphs have been introduced and characterized by Soltan and Chepoi (1983) and Farber and Jamison (1987). CB-graphs and CB-groups generalize systolic (alias bridged) and weakly systolic graphs and groups, which play an important role in geometric group theory. We present metric and local-to-global characterizations of CB-graphs. Namely, we characterize CB-graphs $G$ as graphs whose triangle-pentagonal complexes $X(G)$ are simply connected and balls of radius at most $3$ are convex. Similarly to systolic and weakly systolic graphs, we prove a dismantlability result for CB-graphs $G$: we show that their squares $G^2$ are dismantlable. This implies that the Rips complexes of CB-graphs are contractible. Finally, we adapt and extend the approach of Januszkiewicz and Swiatkowski (2006) for systolic groups and of Chalopin et al. (2020) for Helly groups, to show that the CB-groups are biautomatic.
