Actions of higher rank, irreducible lattices on $\CAT(0)$ cubical complexes

TL;DR

The paper proves that higher rank irreducible lattices in semisimple Lie groups always have a fixed point when acting on CAT(0) cubical complexes, highlighting rigidity properties of such group actions.

Contribution

It establishes a fixed point property for actions of higher rank irreducible lattices on CAT(0) cubical complexes, extending understanding of their geometric rigidity.

Findings

Any action of the lattice on a CAT(0) cubical complex has a fixed point.

The result applies to lattices with $ ext{Q}$-rank $ extgreater= 2$ in semisimple Lie groups.

Supports the conjecture of strong rigidity properties of higher rank lattices.

Abstract

Let $\Gamma$ be an irreducible lattice of $\Q$-rank $\geq 2$ in a semisimple Lie group of noncompact type. We prove that any action of $\Gamma$ on a $\CAT(0)$ cubical complex has a global fixed point.