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

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.