groups over division rings or, later, lattices in the automorphism groups of buildings. The questions we shall investigate are related to group homology, focusing on homological stability, and to the construction of lattices.

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Buildings were discovered by Tits in the fifties as he studied semisimple complex Lie groups in a uniform way using geometry. Later, he generalised the concept of a building to semisimple algebraic groups over arbitrary fields. The first type of buildings to be studied were so-called spherical buildings, which can be thought of as certain simplicial complexes covered by subcomplexes which are tessellated spheres.

Spherical buildings were later classified by Tits in [@Tit:BsT:74]. In higher rank, there is a one-to-one correspondence between spherical buildings and a class of groups consisting (more or less) of semisimple isotropic linear algebraic groups and of isotropic classical groups. The term ‘classical group’ is not rigorously defined, but we take it to mean groups of matrices over division rings preserving certain forms. Typical examples are linear and unitary groups.

Algebraic groups over finite fields, and hence finite spherical buildings play an important role in the classification of finite simple groups. In general, the theory of spherical buildings is very rich and 