The minimal genus problem for right angled Artin groups

TL;DR

This paper studies the minimal genus problem for the second homology of right angled Artin groups, providing bounds, exact cases, and characterizations, with examples showing limitations of realizability.

Contribution

It introduces a lower bound for the minimal genus, characterizes cases where the bound is tight, and explores representability by tori, advancing understanding of homology classes in RAAGs.

Findings

Lower bound equals half the rank of the cap product matrix.

Equality of bound and minimal genus for complete graphs, trees, bipartite graphs.

Counterexample with a pentagon shows not all classes are realizable by tori.

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.