An obstruction to embedding right-angled Artin groups in mapping class groups

TL;DR

The paper demonstrates that certain right-angled Artin groups cannot embed into mapping class groups due to chromatic number obstructions, revealing a new global barrier based on graph coloring properties.

Contribution

It introduces the chromatic number of the graph as a key obstruction to embedding right-angled Artin groups into mapping class groups.

Findings

Existence of right-angled Artin groups of cohomological dimension two that do not embed

Chromatic number of the defining graph limits embeddability into mapping class groups

Chromatic number of the graph is a global obstruction to embedding

Abstract

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\gam$ to embed into the mapping class group of a surface $S$, we show that the chromatic number of $\gam$ cannot exceed the chromatic number of the clique graph of the curve graph $\mathcal{C}(S)$. Thus, the chromatic number of $\gam$ is a global obstruction to embedding the right-angled Artin group $A(\gam)$ into the mapping class group $\Mod(S)$.