Right-angled Artin groups and curve graphs of nonorientable surfaces

TL;DR

This paper demonstrates that right-angled Artin groups associated with certain finite subgraphs of the curve graph of a nonorientable surface can be embedded into its mapping class group, revealing new algebraic structures within these groups.

Contribution

It proves that all right-angled Artin groups on finite full subgraphs of the essential two-sided curve graph embed into the mapping class group of a nonorientable surface, and identifies a graph whose RAAG embeds despite not being a subgraph.

Findings

Every finite full subgraph's RAAG embeds into the mapping class group.

Existence of a graph whose RAAG embeds but is not a subgraph.

Extension of embedding results to nonorientable surfaces.

Abstract

Let $N$ be a closed nonorientable surface with or without marked points. In this paper we prove that, for every finite full subgraph $\Gamma$ of $\mathcal{C}^{\mathrm{two}}(N)$, the right-angled Artin group on $\Gamma$ can be embedded in the mapping class group of $N$. Here, $\mathcal{C}^{\mathrm{two}}(N)$ is the subgraph, induced by essential two-sided simple closed curves in $N$, of the ordinal curve graph $\mathcal{C}(N)$. In addition, we show that there exists a finite graph $\Gamma$ which is not a full subgraph of $\mathcal{C}^{\mathrm{two}}(N)$ for some $N$, but the right-angled Artin group on $\Gamma$ can be embedded in the mapping class group of $N$.