A Note on elementarity of virtual dendro-morphisms for higher rank lattices

TL;DR

This paper proves that for higher rank lattices, all virtual dendro-morphisms are elementary, by constructing a specific unitary representation with certain cohomological properties.

Contribution

It establishes that non-elementary virtual dendro-morphisms cannot exist for higher rank lattices, showing a rigidity result in the context of dendro-morphisms.

Findings

Constructed a unitary representation with no invariant vectors.

Demonstrated non-zero classes in bounded cohomology for the representation.

Proved all virtual dendro-morphisms of higher rank lattices are elementary.

Abstract

Let $\Gamma$ be a discrete countable group and let $(\Omega,\mu)$ be an ergodic standard Borel probability $\Gamma$-space. Given any non-elementary virtual dendro-morphism (that is a measurable cocycle in the automorphism group of a dendrite), we construct a unitary representation $V$ with no invariant vectors such that $\text{H}^2_b(\Gamma;V)$ contains a non-zero class. As a consequence, all virtual dendro-morphisms of a higher rank lattice must be elementary.