Henry Wilton

Surface groups among cubulated groups

Free groups and surface groups are often thought of as cousins. One way to make this precise is through their subgroup structure: they share the property that every subgroup of infinite index is free. In fact, it turns out that this property characterises free and surface groups among cubulated hyperbolic groups, and also among other natural classes including one-relator groups. I will discuss these results and their proofs. The main new tool is the “Whitehead complex” associated to a convex subcomplex of a CAT(0) cube complex. The Whitehead complex generalises the classical Whitehead graph used to study automorphisms of free groups.