Affine Λ-buildings, ultrapowers of Lie groups and Riemannian symmetric spaces: an algebraic proof of the Margulis conjecture

Then Γ acts cocompactly on X and X , and the orbit spaces X/Γ and X /Γ are closed manifolds with contractible universal covers. The only nontrivial homotopy groups are thus the fundamental groups, π1(X/Γ) ∼= Γ ∼= π1(X /Γ). It follows that there is a homotopy equivalence X/Γ X /Γ which can be lifted to a map f : X X . This map f can then be shown to be a quasi-isometry. Recall that a (not necessarily continuous) map f : X −→ X ′ between metric spaces if called an (L,C)-quasiisometry if there exist constants L ≥ 1 and C ≥ 0 such that