Tits Geometry, Arithmetic Groups, and the Proof of a Conjecture of Siegel

Enrico Leuzinger

Abstract: Let $X = G/K$ be a Riemannian symmetric space of noncompact type and of rank $\geq 2$. An irreducible, non-uniform lattice $\Gamma\subset G$ in the isometry group of $X$ is arithmetic and gives rise to a locally symmetric space $V=\Gamma\backslash X$. Let $\pi:X\rightarrow V$ be the canonical projection. Reduction theory for arithmetic groups provides a dissection $V=\coprod_{i=1}^k \pi(X_i)$ with $\pi(X_0)$ compact and such that the restiction of $\pi$ to $X_i$ is injective for each $i$. In this paper we complete reduction theory by focusing on metric properties of the sets $X_i$. We detect subsets $C_i$ of $X_i$ (${\Bbb Q}$--Weyl chambers) such that $\pi_{\mid C_i}$ is an isometry and such that $C_i$ is a net in $X_i$. 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 $V$.

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