Actions on CAT(0) cube complexes are powerful geometric tool for both algebraically decomposing groups and establishing subgroup separability results.  I will describe boundaries associated to hyperbolic and relatively hyperbolic groups. With a focus on (quotients of) free products, I will discuss variations on a boundary criteria of Bergeron—Wise for exhibiting cocompact actions on CAT(0) cube complexes.  I will explain some ideas on how to use these tools to show that most (small-cancellation or random density) quotients of free products preserve residual finiteness. This is based on multiple joint works with subsets of Einstein, Krishna MS, Montee, and Steenbock.

TBA