Poisson boundaries of building lattices and rigidity with hyperbolic-like targets

Abstract

We prove that a Poisson boundary of any regular thick Euclidean building, as well as lattices thereof is the space of chambers at infinity of the building with the harmonic measure class. We then use this result to generalize rigidity results of Guirardel-Horbez-L\'ecureux on morphisms and cocycles from lattices in buildings to groups with negative curvature properties.