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.

翻译：本文研究所有球均为凸集的图以及几何作用于其上的群（我们称之为CB图与CB群）。此类图由Soltan与Chepoi（1983年）以及Farber与Jamison（1987年）引入并刻画。CB图与CB群推广了在几何群论中具有重要作用的收缩图（亦称桥接图）与弱收缩图及其对应的群。我们给出了CB图的度量性质与局部到整体的刻画：具体而言，我们将CB图$G$刻画为满足三角形-五边形复形$X(G)$单连通且半径不超过3的球为凸集的图。类似于收缩图与弱收缩图，我们证明了CB图$G$的可拆解性：即其平方图$G^2$是可拆解的。这意味着CB图的Rips复形是可缩的。最后，我们改编并扩展了Januszkiewicz与Swiatkowski（2006年）关于收缩群以及Chalopin等人（2020年）关于Helly群的方法，证明了CB群具有双自动机性。