Finiteness properties of S-arithmetic subgroups of Chevalley groups in characteristic 0

Abstract

We consider in this thesis S-arithmetic subgroups of certain algebraic matrix groups de ned over Q. The simplest example of such a group is Γ = SLn(Z[1/p]). Each of these groups is of type F∞ by a well-known result of Borel and Serre. On a formal level, this means that there is a K(Γ, 1) complex with finite m-skeleton for every m ∈ N. A nice consequence is that Γ is finitely presented. While the method of Borel and Serre is more algebraic, we give here a new, purely geometric, proof that uses Morse theory. Doing so, we first develop the terminology of a Morse function without critical values greater than a constant r > 0, which is de ned on the product of a Riemannian manifold and a metric space. After that, we deduce some properties from the reduction theory of S-arithmetic groups, which we translate into geometric terms to a space X, on which our group acts canonically. Finally, we construct a real-valued function on that space. We show that this is a Morse function in the sense above. From that we deduce the statement concerning the niteness properties of the group.