The RAAGs on the complement graphs of path graphs in mapping class groups

Abstract

In this article, we determine the function $\ell (S_{g,p})$ such that the right-angled Artin group $G(P_m)$ is embedded in the mapping class group $\mathrm{Mod}(S_{g,p})$ if and only if $m$ is not more than $\ell (S_{g,p})$. Using this function and Birman--Hilden theory, we prove that $\mathrm{Mod}(S_{0,p})$ is virtually embedded in $\mathrm{Mod}(S_{g,0})$ if and only if $p\le 2g+2$.

21 pages, 7 figures