Rank rigidity for Helly graphs

Establish a rank-rigidity theorem, or an appropriate form of rank rigidity, for Helly graphs.

Background

The notes compare Helly geometry with CAT(0) geometry and ask whether analogous rank-rigidity phenomena hold. The authors emphasize that even a suitable conjectural formulation is not currently clear.

References

Do Helly graphs satisfy some form of rank rigidity? Even stating a reasonable conjecture is not obvious.

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” item 12

Let $G$ denote a group acting geometrically on a Helly graph $X$. Then one of the following holds:

— Group actions on injective spaces and Helly graphs  (2307.00414 - Haettel, 2023) in Section 14, “Open questions,” after item 12