** "Geometry and Analysis on Groups" Research Seminar **

**Title:**
Boundary rigidity of CAT (0) cube complexes

**Speaker:**
Merlin Incerti-Medici (University of Zurich)

**Abstract:**
Given a CAT (0) cube complex we ask whether it can be completely understood using only its boundary at infinity. We show that this is the case under very light assumptions. Specifically, we introduce a cross ratio on the Roller boundary of a CAT (0) cube complex and show that any cross ratio preserving bijection between Roller boundaries of two CAT (0) cube complexes induces an isometry between the cube complexes themselves. We end by discussing an application of this result in the proof of a Marked Length Spectrum Rigidity Theorem for CAT (0) cube complexes.