### University of Arkansas Topology Seminar: 3/30/2017

### Speaker: Tim Susse

### Title: Hierarchical hyperbolicity for cube complexes

Hierarchically hyperbolic spaces were introduced by Behrstock−Hagen−Sisto to draw connections between the mapping class group and other types of spaces. In this talk we will prove that if *X* a cocompact CAT(0) cube complex, then *X* is hierarchically hyperbolic. We will go over some of the intuition behind HHS's, as well as some basic properties of CAT(0) cube complexes. Then, we will study the geometry of *X* by looking at the collection formed by closing the set of hyperplanes under closest point projection. Special emphasis will be placed on applications of the theorem to the structure of *X*.

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