Continuous Stability Conditions of Type 𝔸 and Measured Laminations of the Hyperbolic Plane

We introduce stability conditions (in the sense of King) for representable modules of continuous quivers of type 𝔸 along with a special criteria called the four point condition. The stability conditions are defined using a generalization of δ functions, called half- δ functions. We show that for a continuous quiver of type 𝔸 with finitely many sinks and sources, the stability conditions satisfying the four point condition are in bijection with measured laminations of the hyperbolic plane. Along the way, we extend an earlier result by the first author and Todorov regarding continuous cluster categories for linear continuous quivers of type 𝔸 and laminations of the hyperbolic plane to all continuous quivers of type 𝔸 with finitely many sinks and sources. We also give a formula for the continuous cluster character.