Graphs with convex balls

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 (Kibernetika 6: 14–18, 1983) and Farber and Jamison (Discrete Math 66: 231–247, 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 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 $$G^2$$ 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. (arXiv preprint arXiv:2002.06895 , 2020) for Helly groups, to show that the CB-groups are biautomatic.