On commuting elements and embeddings of graph groups and monoids

TL;DR

This paper investigates the commutation structures within right-angled Artin groups and trace monoids, revealing restrictions on subgroup embeddings based on graph properties, notably excluding certain four-cycle configurations.

Contribution

It establishes new limitations on the commutation patterns in these groups and monoids, linking graph structure to algebraic subgroup properties.

Findings

Groups with certain graph structures lack four-element commutation cycles

No subgroup isomorphic to a product of non-abelian free groups exists under these conditions

Results extend to trace monoids with similar properties

Abstract

We study commutation properties of subsets of right-angled Artin groups and trace monoids. We show that if Gamma is any graph not containing a four-cycle without chords, then the group G(Gamma) does not contain four elements whose commutation graph is a four-cycle; a consequence is that G(Gamma) does not have a subgroup isomorphic to a direct product of non-abelian free groups. We also obtain corresponding and more general results for monoids.