The study of groups acting on spaces satisfying various nonpositive
curvature conditions is an area of significant interest in geometric
group theory. Diverse families of groups have been studied in this
context, including CAT(0) groups, cubical groups, hierarchically
hyperbolic groups, Helly groups, systolic groups, quadric groups, etc.
In this talk I will discuss the strong shortcut property, a weak
nonpositive curvature condition of rough geodesic metric spaces that
unifies all of these families of groups and which also includes the
Heisenberg group. I will give an asymptotic cone characterization of
the strong shortcut property and discuss some group theoretic
consequences of nice actions on strong shortcut spaces.