Buildings, Group Homology and Lattices: the Construction of Wagoner Complexes, General Homological Stability for Classical Groups and a New Construction of Lattices on Affine Buildings

This thesis addresses questions concerning the homology of groups related to buildings. We also give a new construction of lattices in such groups and investigate their group homology. Specifically, we construct a simplicial complex called the Wagoner complex associated to any group of Kac-Moody type. For a 2-spherical group G of Kac-Moody type, we show that the fundamental group of the Wagoner complex is almost always isomorphic to the Schur multiplier of the little projective group of G. Furthermore, we present a general method to prove homological stability for groups acting strongly transitively on weak spherical buildings. We use this method to prove strong homological stability results for special linear groups over infinite fields and for unitary groups over division rings, improving the previously best known results in many cases. Finally, we give a new construction method for buildings of types ~A2 and ~C2 with cocompact lattices in their automorphism groups, yielding very explicit presentations of the lattices as well as very explicit descriptions of the buildings. To the author's knowledge, these are the first known presentations of lattices in buildings of type ~C2.