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

Creators: Day, Matthew B.

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(2g, Z). 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 Γ is defined to be a subgroup of the automorphism group of the right-angled Artin group AΓ of Γ. We also prove that the kernel of AutAΓ → AutH_1(AΓ) is finitely generated, generalizing a theorem of Magnus.

Additional Information

© 2009 Mathematical Sciences Publishers. Received: 31 July 2008; accepted: 25 November 2008; published: 8 January 2009. The results of this paper originally appeared in my Ph.D. thesis at the University of Chicago. Some of the research was done under the support of a graduate research fellowship from the National Science Foundation, and the paper was prepared under the support of NSF postdoctoral fellowship DMS-0802918. I am deeply grateful to Benson Farb, my thesis advisor, for many useful conversations and comments on earlier versions of this work. I am grateful to Karen Vogtmann for conversations about this project and to Ruth Charney for conversations and for helping me find an obscure reference. I would also like to thank Hanna Bennett, Nathan Broaddus, Thomas Church, Jim Fowler and Benjamin Schmidt for comments on earlier versions of this paper. Mathematical Subject Classification: Primary: 20F28, 20F36

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