具有凸球的图
Graphs with convex balls
AI总结：
本文研究所有球均为凸集的CB-图与CB-群，给出其度量与局部到整体刻画，证明CB-图的平方可拆解、Rips复形可缩，并证明CB-群是双自动的。
摘要：
本文研究所有球均为凸集的图，以及几何作用于这些图的群（我们称之为CB-图和CB-群）。这些图由Soltan和Chepoi（1983）以及Farber和Jamison（1987）引入并刻画。CB-图和CB-群推广了systolic（又称bridged）图和群以及弱systolic图和群，它们在几何群论中具有重要作用。我们给出了CB-图的度量刻画和局部到整体刻画。具体而言，我们将CB-图G刻画为其三角-五角复形X(G)单连通且半径至多为3的球均为凸集的图。与systolic图和弱systolic图类似，我们证明了CB-图G的一个可拆解性结果：我们证明了它们的平方G^2是可拆解的。这意味着CB-图的Rips复形是可缩的。最后，我们改造并推广了Januszkiewicz和Swiatkowski（2006）针对systolic群的方法以及Chalopin等人（2020）针对Helly群的方法，证明了CB-群是双自动的。
英文摘要：
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.
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