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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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,). 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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800219

Digital Object Identifier
doi:10.2140/gt.2009.13.857

Mathematical Reviews number (MathSciNet)
MR2470965

Zentralblatt MATH identifier
1181.20032

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

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

Citation

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. https://projecteuclid.org/euclid.gt/1513800219


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References

  • M Bestvina, K-U Bux, D Margalit, Dimension of the Torelli group for ${\rm Out}(F\sb n)$, Invent. Math. 170 (2007) 1–32
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer, New York (1982)
  • R Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007) 141–158
  • R Charney, J Crisp, K Vogtmann, Automorphisms of $2$–dimensional right-angled Artin groups, Geom. Topol. 11 (2007) 2227–2264
  • R Charney, K Vogtmann, Finiteness properties of automorphism groups of right-angled Artin groups, to appear in Bull. London. Math. Soc. Available at \setbox0\makeatletter\@url http://people.brandeis.edu/~charney/webpubs.htm {\unhbox0
  • M B Day, Peak reduction and presentations for automorphism groups of right-angled Artin groups, Geom. Topol. 13 (2009) 817–855
  • B Farb, D Margalit, A primer on mapping class groups, Book draft Available at \setbox0\makeatletter\@url http://www.math.utah.edu/~margalit/primer/ {\unhbox0
  • N Kawazumi, Cohomological aspects of Magnus expansions
  • M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. $(2)$ 52 (1995) 318–334
  • R C Lyndon, P E Schupp, Combinatorial group theory, Classics in Math., Springer, Berlin (2001) Reprint of the 1977 edition
  • W Magnus, Über $n$–dimensionale Gittertransformationen, Acta Math. 64 (1935) 353–367
  • W Magnus, A Karrass, D Solitar, Combinatorial group theory. Presentations of groups in terms of generators and relations, second edition, Dover, Mineola, NY (2004)
  • J McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975) 205–213
  • J McCool, Generating the mapping class group (an algebraic approach), Publ. Mat. 40 (1996) 457–468
  • R C Penner, The moduli space of a punctured surface and perturbative series, Bull. Amer. Math. Soc. $($N.S.$)$ 15 (1986) 73–77
  • A Pettet, The Johnson homomorphism and the second cohomology of ${\rm IA}\sb n$, Algebr. Geom. Topol. 5 (2005) 725–740
  • J-P Serre, Lie algebras and Lie groups, Lecture Notes in Math. 1500, Springer, Berlin (2006) 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) edition
  • H Servatius, Automorphisms of graph groups, J. Algebra 126 (1989) 34–60
  • L VanWyk, Graph groups are biautomatic, J. Pure Appl. Algebra 94 (1994) 341–352
  • H Zieschang, Über Automorphismen ebener diskontinuierlicher Gruppen, Math. Ann. 166 (1966) 148–167