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G&T Monographs |
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Symplectic structures on right-angled Artin groups: Between
the mapping class group and the symplectic group
Matthew B Day
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Geometry & Topology 13 (2009)
857–899
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Abstract
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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 → AutH1(AΓ) is finitely generated, generalizing a theorem of
Magnus.
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Keywords
peak reduction, symplectic structure,
finite generation, right-angled Artin group
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Mathematical Subject Classification
Primary: 20F28, 20F36
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Publication
Received: 31 July 2008
Accepted: 25 November 2008
Published: 8 January 2009
Proposed: Joan Birman
Seconded: Leonid Polterovich, Ron Stern
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