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, the minimal genus of a second homology class of a RAAG is not always realised by a disjoint union of tori as an example we construct in the pentagon shows.
Cite
@article{arxiv.2108.02914,
title = {The minimal genus problem for right angled Artin groups},
author = {Rachael Boyd and Thorben Kastenholz and Jean Pierre Mutanguha},
journal= {arXiv preprint arXiv:2108.02914},
year = {2023}
}
Comments
v1: 19 pages, 4 figures; comments welcome. v2: 20 pages; minor changes to incorporate referee comments