In 2013 Prof. Dr. Bux, Prof. Dr. Köhl and Dr. Witzel published an article on higher finiteness properties of reductive arithmetic groups in positive characteristic ([BKW13]). An essential tool in their work is the transmission of algebraic reduction theory into pure geometry:

Given a semisimple linear algebraic group G and a finite product of local function fields k_f, they created a reduction theory on the building accociated to G(k_f). Later on Prof. Dr. Bux asked, whether they created the right kind of geometric reduction theory and if there is a universally valid reduction theory on arbitrary CAT(0)-spaces. As an intermediate step reaching this highly ambitious aim he asked for a second example:

In this work we create an analogous reduction theory on the product of a symmetric space and a building associated to G(k_n), where k_n is a finite product of local number fields.