Cores and intersection numbers of group cubulations
Mark Hagen (University of Cambridge)
Wednesday 26th April, 2017 16:00-17:00 tbc
A (multi)curve on a surface gives an easy-to-visualise splitting of a group (in this case, the fundamental group of the surface) as a graph of groups. Bass-Serre theory tells us that group splittings correspond to actions on trees. Many groups admit a multitude of actions on trees, so it is natural to compare different actions of a single group on multiple trees. In the 1990s, Scott expanded the notion of the intersection number of a pair of curves to account for pairs of splittings of an arbitrary group; two splittings have intersection number 0 when they can be combined into a single splitting. Later, Guirardel gave a geometric version, by examining the product of the two tree actions and building the "core", a well-behaved invariant subspace of the product of the two trees, whose covolume coincides with the intersection number. In this talk, I'll discuss a new construction that generalises the Guirardel core in two ways: instead of two actions, we work with any finite number, and instead of actions on trees, we work with actions on CAT(0) cube complexes. I will give some background on CAT(0) cube complexes, and why they are "high-dimensional trees". I'll then discuss how to use the generalised core to give a new solution of the Nielsen realisation problem for finite groups of outer automorphisms of a free group. This talk is based on ongoing joint work with Henry Wilton.