On Isometric Extension of Moebius Maps

Details On Isometric Extension of Moebius Maps On Isometric Extension of Moebius Maps

2012-03-28

arxiv.org/abs/1203.6212v1

Mathematics

Dynamical Systems

Scientific paper

We consider Moebius homeomorphisms $f : \partial X \to \partial Y$ between boundaries of CAT(-1) spaces $X,Y$ equipped with visual metrics. We prove that if $X,Y$ are proper and geodesically complete then $f$ extends to a $(1, \log 2)$-quasi-isometry with image $1/2\log 2$-dense in $Y$. We prove that if $X,Y$ are in addition metric trees then $f$ extends to a surjective isometry. The proofs involve a study of a space $\mathcal{M}(\partial X)$ of metrics on $\partial X$ Moebius equivalent to a visual metric and an isometric embedding of $X$ into $\mathcal{M}(\partial X)$.

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