#### Volume 13, issue 2 (2009)

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

### Matthew B Day

Geometry & Topology 13 (2009) 857–899
##### 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 $Sp\left(2g,ℤ\right)$. 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.

##### Keywords
peak reduction, symplectic structure, finite generation, right-angled Artin group
##### Mathematical Subject Classification 2000
Primary: 20F36, 20F28