Symplectic structures on right-angled Artin groups: Between the mapping class group and the symplectic group

Abstract

We define a family of groups that include the mapping class group of a genus $g$ surface with one boundary component and the integral symplectic group $Sp\left(2g,\mathbb{Z}\right)$. We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with $2g$ vertices. The mapping class group over the graph $\Gamma$ is defined to be a subgroup of the automorphism group of the right-angled Artin group $A_{\Gamma }$ of $\Gamma$. We also prove that the kernel of $Aut{A}_{\Gamma }\to Aut{H}_1\left(A_{\Gamma }\right)$ is finitely generated, generalizing a theorem of Magnus.