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

Day, Matthew B¹

¹ Department of Mathematics, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91101, USA

Geometry & topology, Tome 13 (2009) no. 2, pp. 857-899.

Résumé

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.

Détail

DOI : 10.2140/gt.2009.13.857

Keywords: peak reduction, symplectic structure, finite generation, right-angled Artin group

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