The rigidity of lattices in products of trees

Each complete CAT(0) space has an associated topological space, called visual boundary that coincides with the Gromov boundary in case that the space is hyperbolic. A CAT(0) group $G$ is called boundary rigid if the visual boundaries of all CAT(0) spaces admitting a geometric action by $G$ are homeomorphic. If $G$ is hyperbolic, $G$ is boundary rigid. If $G$ is not hyperbolic, $G$ is not always boundary rigid. The first such example was found by Croke-Kleiner.

In this talk we will see that every group acting freely and cocompactly on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the boundary homeomorphic to a join of two copies of the Cantor set. The proof of this result uses a generalization of classical dynamics on boundaries introduced by Guralnik and Swenson. I will explain the idea of this generalization by explaining a higher-dimensional version of classical North-south-dynamics obtained this way.