The minimal genus problem for right angled Artin groups

Rachael Boyd, Thorben Kastenholz, Jean Pierre Mutanguha

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Article number93
JournalGeometriae Dedicata
Volume217
Issue number5
DOIs
StatePublished - Oct 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Geometry and Topology

Keywords

  • Group homology
  • Minimal genus
  • Right angled Artin groups

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