The minimal genus problem for right angled Artin groups

Abstract

We investigate the minimal genus problem for the second homology of a right angled Artin group (RAAG). Firstly, we present a lower bound for the minimal genus of a second homology class, equal to half the rank of the corresponding cap product matrix. We show that for complete graphs, trees, and complete bipartite graphs, this bound is an equality, and furthermore in these cases the minimal genus can always be realised by a disjoint union of tori. Additionally, we give a full characterisation of classes that are representable by a single torus. However, it is not true in general that the minimal genus of a second homology class of a RAAG is necessarily realised by a disjoint union of tori: we construct a genus two representative for a class in the pentagon RAAG.