Tits geometry, arithmetic groups, and the proof of a conjecture of

Abstract. Let X = G/K be a Riemannian symmetric space of non-compact type and of rank ≥ 2. An irreducible, non-uniform lattice Γ ⊂ G in the isometry group of X is arithmetic and gives rise to a locally symmet-ric space V = Γ\X. Let pi: X → V be the canonical projection. Reduction theory for arithmetic groups provides a dissection V = ∐k i=1 pi(Xi) with pi(X0) compact and such that the restiction of pi to Xi is injective for each i. In this paper we complete reduction theory by focusing on metric prop-erties of the sets Xi. We detect subsets Ci of Xi (Q–Weyl chambers) such that pi|Ci is an isometry and such that Ci is a net in Xi. This result is then used to prove a conjecture of C.L. Siegel. We also show that V is quasi-isometric to the Euclidean cone over a finite simplicial complex and study the Tits geometry of V