# PREPRINT 10/1996

## CATCHING GEODESICS IN HADAMARD SPACES

This preprint was accepted October 1996.

ABSTRACT:
For every quasi-isometric map $f:X\to Y$
of Hadamard spaces we define its asymptotic limit $s_f$
which sends the boundary at infinity $\bin X$ to the cone $\Cin Y$
over $\bin Y$ and establish its analytical properties. In the case when
$X$ and $Y$ are cocompact rank one spaces with respect to the same discrete
isometry group $\Ga$ and hence $\Ga$-equivariantly quasi-isometric
we give a sufficient condition for $s_f$ to be an equivariant homeomorphism
between $\bin X$ and $\bin Y$ w.r.t. the standard topologies and
biLipschitz homeomorphism w.r.t. Tits metrics. Apart of this condition there
is a large number of equivariantly quasi-isometric cocompact Hadamard spaces
whose boundaries at infinity are not equivariantly homeomorphic. This answers
a question of M. Gromov.



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