This is joint work with Inna Cadeboscq (Warwick). Profinite groups are compact totally disconnected groups, or equivalently projective limits of finite groups. This class of groups appears naturally in infinite Galois theory, but they can be studied for their own sake (which will be the case in this talk). We are interested in pro-$p$ groups, i.e. projective limits of finite $p$-groups. For instance, the group $SL(n , \mathbb{Z}_p)$ - and in general any maximal compact subgroup in a Lie group over a local field of residual characteristic $p$ - contains a pro-$p$ group of finite index. The latter groups can be seen as pro-$p$ Sylow subgroups in this situation (they are all conjugate by a non-positive curvature argument).
We will present an a priori non-linear generalization of these examples, arising via automorphism groups of spaces that we will gently introduce: buildings. The main result is the existence of a wide class of automorphism groups of buildings which are simple and whose maximal compact subgroups are virtually finitely generated pro-$p$ groups. This is only the beginning of the study of these groups, where the main questions deal with linearity, and other homology groups.
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