Genus bounds in right-angled Artin groups

TL;DR

This paper establishes lower bounds on stable commutator length in right-angled Artin groups based on the chromatic number and triangle-free conditions of their defining graphs, using geometric methods.

Contribution

It provides new bounds on stable commutator length in right-angled Artin groups depending on graph properties, extending previous geometric approaches.

Findings

Stable commutator length ≥ 1/(6k) for graphs with chromatic number k.

Stable commutator length ≥ 1/20 for triangle-free graphs.

Results derived via elementary geometric arguments.

Abstract

We show that in any right-angled Artin group whose defining graph has chromatic number $k$, every non-trivial element has stable commutator length at least $1/(6k)$. Secondly, if the defining graph does not contain triangles, then every non-trivial element has stable commutator length at least $1/20$. These results are obtained via an elementary geometric argument based on earlier work of Culler.