Curtain Model for CAT(0) Spaces and Isometries

Abstract: This paper studies the dynamics of isometries in the curtain model, which is used to capture the hyperbolicity in a fixed CAT(0) space. We establish several fundamental properties, fully classify the behavior of semisimple isometries of a CAT(0) space in the associated curtain model, and prove that an isometry is contracting in a CAT(0) space if and only if it becomes loxodromic in the curtain model. Additionally, we exclude the presence of parabolic actions in most cases of interest, allowing the use of ping-pong like techniques on the curtain model to provide insights into the study of CAT(0) groups.