CAT(0) cube complexes were introduced by Gromov merely as examples of metric spaces of non-positive curvature, but now they play a prominent role in geometric group theory. One reason for this is that many interesting groups are known to act nicely on these spaces, including free and surface groups, small cancellation groups, 1-relator groups with torsion, and many 3 manifold groups. Another reason is that some of these groups are, in addition, virtually special, notion defined by Haglund and Wise that implies being (up to finite index) the subgroup of some right-angled Artin group.
In the first lecture, we will define CAT(0) cube complexes, explore some of their combinatorial structure, and discuss some examples of cubulated groups. For the second lecture, we will introduce the class of virtually special groups, review some of their properties, and mention some criteria for virtual specialness. We will end the mini-course with a discussion of the main techniques for studying cubulated hyperbolic groups, focusing on some theorems of Wise and Agol. If time permits, I will mention a few things about the relatively hyperbolic case.
These videos were part of the Geometric group theory without boundaries II virtual summer school.
