Constructing equivariant maps for representations

Original · EN

We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic k-space Hᵏ, and if R is a representation of G into the group of the isometries of Hⁿ, then any R-equivariant map F from Hᵏ to Hⁿ 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. Moreover, under an additional hypothesis, we show that the weak extension we obtain is actually a measurable R-equivariant map from the boundary of Hᵏ to the closure of Hⁿ. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves.