Rank rigidity for CAT(0) cube complexes

TL;DR

This paper proves the Rank Rigidity Conjecture for CAT(0) cube complexes, showing that groups acting on these spaces contain rank one isometries, with implications for geometric group theory and the structure of such complexes.

Contribution

It establishes the Rank Rigidity Conjecture for CAT(0) cube complexes and derives several geometric and algebraic consequences, including a new proof of the Tits Alternative.

Findings

Groups acting on CAT(0) cube complexes contain rank one isometries.

A geometric proof of the Tits Alternative is provided.

Characterization of products of trees via bounded cohomology.

Abstract

We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.