Graphs with convex balls

TL;DR

This paper studies graphs where all balls are convex, providing characterizations, properties, and showing that associated groups are biautomatic, thus extending understanding of geometric and algebraic properties of these graphs.

Contribution

It introduces metric and local-to-global characterizations of CB-graphs, proves dismantlability of their squares, and shows CB-groups are biautomatic, extending prior work on systolic and Helly groups.

Findings

CB-graphs are characterized by simply connected triangle-pentagonal complexes and convex balls of radius up to 3.

Squares of CB-graphs are dismantlable, implying their Rips complexes are contractible.

CB-groups are biautomatic, extending properties known for systolic and Helly groups.

Abstract

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.