A group is coherent if all its finitely generated subgroups are finitely presented. Wise proposes a bold classification of coherent two dimensional groups: he conjectures that G is coherent if and only if G is virtually the fundamental group of a 2-complex with nonpositive immersions if and only if the second L²-Betti number of G vanishes. In this talk we will define the concepts appearing in Wise's Conjecture, and we will see that the conjecture holds in the class of virtually special groups. We will also discuss applications to coherence of Coxeter groups and random groups, as well similar results for higher-dimensional coherent groups. This is based on joint work with Pablo Sánchez-Peralta.
