Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings

for all x ∈ X . Quasi-isometries occur naturally in the study of the geometry of discrete groups since the length spaces on which a given finitely generated group acts cocompactly and properly discontinuously by isometries are quasi-isometric to one another [Gro]. Quasi-isometries also play a crucial role in Mostow’s proof of his rigidity theorem: the theorem is proved by showing that equivariant quasi-isometries are within bounded distance of isometries. This paper is concerned with the structure of quasi-isometries between products of symmetric spaces and Euclidean buildings. We recall that Euclidean space, hyperbolic space, and complex hyperbolic space each admit an abundance of self-quasi-isometries [Pan]. For example we get quasiisometries E −→ E by taking shears in rectangular (x1, x2) 7→ (x1, x2 + f(x1)) or polar (r, θ) 7→ (r, θ+ f(r) r ) coordinates, where f : R −→ R and g : [0,∞) −→ R are Lipschitz. Any diffeomorphism Φ : ∂H −→ ∂H of the ideal boundary can be extended continuously to a quasi-isometry Φ : H −→ H . Likewise any contact diffeomorphism ∂Φ : ∂CH −→ ∂CH) can be extended continuously to a quasi-isometry Φ : CH −→ CH) [Pan]. Quasi-isometries of the remaining rank 1 symmetric spaces of noncompact type, on the other hand, are very special. They are essentially isometries: