In this talk we will discuss a generalisation of Sageev's wallspace construction that allows to improve geometrical properties of a space by looking at its dual with respect to a set of walls. Specifically , we'll look at the interactions with hyperbolicity and focus on two applications. In CAT(0) spaces, these techniques allow to construct a "universal hyperbolic quotient", called the curtain model, that is analogous to the curve graph of a surface. When focusing on a space that is already hyperbolic, we the construction can be used to improve its fine properties, and in particular we address a conjecture of Rips and Sela and show that residually finite hyperbolic groups admit globally stable cylinders. This is joint work with Petyt and Zalloum.
