Quasi-isometric Rigidity of Nonuniform Lattices in Higher Rank Symmetric Spaces

Thus φ is bi-Lipschitz on large scales. If there exists a constant C ′ such that every point of Y is within a distance C′ of a point of φ(X), then there is a quasi-isometric embedding ψ : Y → X which is a “coarse inverse of φ”, i.e. supx∈X dX(x, ψ(φ(x))) <∞ and supy∈Y dY (y, φ(ψ(y))) < ∞. In this case φ is called a quasi-isometry and the spaces X and Y are said to be quasi-isometric. A basic example to keep in mind is that the fundamental group π1(M) (endowed with the word metric) of a compact Riemannian manifold M is quasi-isometric to the universal cover M̃ of M . In this paper we prove the following theorem: