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

Let $\mathrm{\Gamma }$ be a discrete countable group and let $(\mathrm{\Omega },\mu )$ be an ergodic standard Borel probability $\mathrm{\Gamma }$-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 ${\text{H}}_b^2(\mathrm{\Gamma };V)$ contains a non-zero class. As a consequence, all virtual dendro-morphisms of a higher rank lattice must be elementary.