CAT(0) cube complexes (CCCs) may be seen as generalizations of trees. These complexes while highly structured are also easily constructed, making them a playground for discovery. However, not all groups admit nontrivial actions on them!

The Furstenberg-Poisson (FP) boundary associated to a generating random walk on a discrete countable group always exists as an abstract measure theoretic object. Therefore, it is desirable to realize the FP boundary as a geometric boundary of some kind, allowing us to connect geometric and analytic information about the group. 

In this mini-course, we will explore connections between CCCs and the FP boundaries of groups acting on them. Among other things we will prove: The FP boundary of a group acting on a finite dimensional CCC always admits a virtually-equivariant measurable map to the Roller Compactification. Moreover,  under weak conditions, the Roller boundary is a model for the Furstenberg-Poisson boundary.

This tour will take us through various topics including Bader-Furman's Isometric Ergodicity, Kaimanovich's Strip Condition, and intervals associated to probability measures on CCCs.

Lecture Materials: 
https://sites.google.com/view/talialovesmath/media/cat0-cube-complexes-and-fp-boundary