Abstract:

In a sequence of papers, D. Wise (together with F. Haglund, T. Hsu and others) studied a special class of non-positively curved cube complexes, that they call special cube complexes. These are required to satisfy some transversality conditions, and as a result their fundamental groups turned out to have some surprising properties, e.g. they are linear (over Z), can be embedded in right angled Artin groups, as well as some strong implications for their profinite topology.

Recently, Wise has managed to apply these special cube complexes, to answer affirmatively some long standing questions (conjectures) in 3-manifold topology.

We intend to explain some of these implications, as well as some of the basic constructions. No previous knowledge will be assumed, apart from basic topology.