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

Abstract

Let Γ \Gamma be a discrete countable group and let ( Ω , μ ) (\Omega ,\mu ) be an ergodic standard Borel probability Γ \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 V with no invariant vectors such that H b 2 ⁡ ( Γ ; V ) \operatorname {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.