Geometry & Topology

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

Matthew B Day

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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,). 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.

Article information

Geom. Topol., Volume 13, Number 2 (2009), 857-899.

Received: 31 July 2008
Accepted: 25 November 2008
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups 20F28: Automorphism groups of groups [See also 20E36]

peak reduction symplectic structure finite generation right-angled Artin group


Day, Matthew B. Symplectic structures on right-angled Artin groups: Between the mapping class group and the symplectic group. Geom. Topol. 13 (2009), no. 2, 857--899. doi:10.2140/gt.2009.13.857.

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