The boundary rigidity of lattices in products of trees

TL;DR

This paper proves that groups acting freely and vertex-transitively on a product of two regular trees are boundary rigid, meaning their associated CAT(0) spaces have visual boundaries homeomorphic to a join of two Cantor sets.

Contribution

It establishes boundary rigidity for groups acting on products of two regular trees, linking geometric actions to boundary topology in CAT(0) spaces.

Findings

Groups acting freely and vertex-transitively on product of two trees are boundary rigid.

Visual boundary of associated CAT(0) space is homeomorphic to a join of two Cantor sets.

Boundary rigidity holds for all such groups regardless of specific actions.

Abstract

We show that every group acting freely and vertex-transitively by isometries 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 visual boundary homeomorphic to a join of two copies of the Cantor set.