Sageev's cube complex dual to a collection of curves on a hyperbolic surface

Jonah Gaster (McGill University)

Sageev gave a very general construction of a CAT(0) cube complex dual to a `space with walls', and this construction has proved extraordinarily useful in recent celebrated work of Agol, Wise, and others. In one of the simplest nontrivial settings, this construction produces a non-positively curved cube complex dual to a finite collection of non-homotopic essential closed curves on a surface. I will describe Sageev's construction in this setting, discuss some related geometric / combinatorial data, and give an application to the analysis of the length function associated to a curve.