Constructing equivariant maps for representations

We show that if Γ is a discrete subgroup of the group of the isometries of ℍ k , and if ρ is a representation of Γ into the group of the isometries of ℍ n , then any ρ-equivariant map F:ℍ k →ℍ n extends to the boundary in a weak sense in the setting of Borel measures. As a consequence of this fact, we obtain an extension of a result of Besson, Courtois and Gallot about the existence of volume non-increasing, equivariant maps. Then, we show that the weak extension we obtain is actually a measurable ρ-equivariant map in the classical sense. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves. For example, we prove that if Γ is of divergence type and ρ is non-elementary, then there exists a measurable map D:∂ℍ k →∂ℍ n conjugating the actions of Γ and ρ(Γ). Related applications are discussed.