CAT(0) cube complexes are charming objects with many striking properties. For example, they admit two interesting, and naturally coupled metrics: the CAT(0) metric and the median metric, allowing one to access the rich tools from each of those worlds. The study of low dimensional cohomology of a group touches upon several important aspects of group theory: Property (T), the Haagerup Property, stable commutator length, and even superrigidity. In this talk, we will discuss CAT(0) cube complexes, and how they provide a nice framework for finding low dimensional cohomology classes such as the Haagerup Cocycle and various generalization of the Brooks cocycle.

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