Abstract

Let G be a reductive algebraic group over a global function field K with a residue field of order q, let A_K be the ring of adeles over K, and let P be a maximal K-parabolic subgroup of G. Harder defines the distance of a maximal compact subgroup C of G(A_K) from P as the volume of the intersection of C and P(A_K) with respect to the Tamagawa measure.

For each finite non-empty set S of places of K, the logarithm of this distance function to the basis q yields a Busemann function on the product of the affine buildings of G(K_s), where s runs through S.

The family of all these Busemann functions, as P varies through the maximal K-parabolic subgroups of G, allows one to define a Morse function on this product of affine buildings, which can be used to derive the finiteness properties of the S-arithmetic subgroup of G(K).

In my talk I will state Harder’s fundamental theorem of reduction theory, derive the Busemann functions and the Morse function, and explain how this setup can be used in order to establish finiteness properties of S-arithmetic groups.