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:在本文中,我们调查了所有球都是混凝土的图表和几何组(我们称之为CB-graphs和CB-组)。这些图表被采用,并以Soltan和Chepoi(1983年)以及Farber和Jamison(1987年)为特征。CB-graphs和CB组一般化了Systolic (alips bild) 和薄弱的Systolic 图表和组,这些组在几何组理论中起着重要作用。我们展示了CBB-graph的度和地方到全球特征。也就是说,我们把CB-graphes $G$-graphs(G$X)(1983年)和Farber和Jamison(1987年)。Camison(1987年)。CBC-graphy-graphs)和CBystlicol 图表与Systemal Agrouple 和Shecliversal Cal-CBals 集团(20 Chabal-CBal) 和Shecal-CBal-CBIal-CBIal 集团(20 和Sy-CBIal-C-C-C-C-C-C-C-CBaltral)的缩合。我们证明 和CB-CB-CB-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C-C