Groups quasi-isometric to symmetric spaces

Abstract

AbstractWe determine thestructureoffinitely generated groupswhich arequasi-isometricto symmetric spaces of noncompact type, allowing Euclidean de Rham factorsIf X is a symmetric space of noncompact type with no Euclidean de Rham fac-tor, and Γ is a finitely generated group quasi-isometric to the product E k ×X,then there is an exact sequence 1 → H → Γ → L → 1 where H contains afinite index copy of Z k and L is a uniform lattice in the isometry group of X. 1 1 Introduction The main result of this paper is the following theorem.Theorem 1.1 Let X be a symmetric space of noncompact type with no Euclidean deRham factor, and let Nil be a simply connected nilpotent Lie group equipped with aleft-invariant Riemannian metric. Suppose that Γ is a finitely generated group quasi-isometric to Nil× X . Then there is an exact sequence 1 −→ H−→ Γ −→ L−→ 1 (1) where H is a finitely generated group quasi-isometric to Nil and L is a uniform latticein the isometry group of X . In particular, when Nilis the trivial group then Γ is a finite extension of a uniformlattice in Isom(X), and when Nil≃ R