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

TL;DR

This paper characterizes the Poisson boundaries of regular Euclidean buildings and their lattices, and applies these results to extend rigidity theorems for group morphisms and cocycles into negatively curved groups.

Contribution

It establishes the Poisson boundary for Euclidean buildings and lattices, and generalizes rigidity results to groups with negative curvature properties.

Findings

Poisson boundary of Euclidean buildings is the space of chambers at infinity

Lattices in buildings share this Poisson boundary with the building itself

Rigidity results are extended to groups with negative curvature

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.