arXiv math/04050282004-05math.GTmath.DS

Constructing equivariant maps for representations

S. Francaviglia

备注：: Changes from V1: The paper has been substantially reorganised. New applications added. This is the version accepted for pubblication. To appear on Ann. Inst. Fourier

英文摘要：

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 H^n, then any R-equivariant map F from H^k to 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. 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 H^n. We use this fact to obtain measurable versions of Cannon-Thurston-type results for equivariant Peano curves.

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