---
author:
- Jan Essert
bibliography:
- '../biblio.bib'
title: 'Buildings, Group Homology and Lattices'
---

[**Abstract**]{}

This thesis discusses 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 with weak spherical Tits systems*, that is, 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 $\tilde A_2$ and $\tilde C_2$ with *cocompact lattices* in their full automorphism groups. Almost all of these buildings are exotic in the case $\tilde C_2$. In the $\tilde A_2$-case, it is not known whether these 