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

Day, Matthew B

doi:10.2140/gt.2009.13.857

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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

DOI: 10.2140/gt.2009.13.857

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 $Sp (2 g, ℤ)$ . 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 $2 g$ 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 $A_{Γ}$ of $Γ$ . We also prove that the kernel of $Aut A_{Γ} \to Aut H_{1} (A_{Γ})$ 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

References

Publication

Received: 31 July 2008

Accepted: 25 November 2008

Published: 8 January 2009

Proposed: Joan Birman

Seconded: Leonid Polterovich, Ron Stern

Authors

Matthew B Day
Department of Mathematics California Institute of Technology 1200 E California Blvd Pasadena, CA 91101 USA
http://www.its.caltech.edu/~mattday

Volume 13, issue 2 (2009)