Constructing equivariant maps for representations

2004/05/03 by Stefano Francaviglia, Francaviglia, S.
Mathematics · #37A99 (Secondary) #57M50 (Primary) #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.math/0405028

openalex publication_date 2004/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if G is a discrete subgroup of the group of the isometries of the hyperbolic k-space H^k, and if R is a representation of G into the group of the isometries of Hⁿ, then any R-equivariant map F from H^k 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 H^k to the closure of Hⁿ. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves.

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