Geometry & Topology

Peak reduction and finite presentations for automorphism groups of right-angled Artin groups

Matthew B Day

Abstract

We generalize the peak reduction algorithm (Whitehead’s theorem) for free groups to a theorem about a general right-angled Artin group $AΓ$. As an application, we find a finite presentation for the automorphism group $AutAΓ$ that generalizes McCool’s presentation for the automorphism group of a finite rank free group. We also consider a stronger generalization of peak reduction, giving a counterexample and proving a special case.

Article information

Source
Geom. Topol., Volume 13, Number 2 (2009), 817-855.

Dates
Revised: 4 December 2008
Accepted: 22 November 2008
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.gt/1513800218

Digital Object Identifier
doi:10.2140/gt.2009.13.817

Mathematical Reviews number (MathSciNet)
MR2470964

Zentralblatt MATH identifier
1226.20024

Citation

Day, Matthew B. Peak reduction and finite presentations for automorphism groups of right-angled Artin groups. Geom. Topol. 13 (2009), no. 2, 817--855. doi:10.2140/gt.2009.13.817. https://projecteuclid.org/euclid.gt/1513800218

References

• K-U Bux, R Charney, K Vogtmann, Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups, to appear in Groups Geom. Dyn. Available at \setbox0\makeatletter\@url http://people.brandeis.edu/~charney/webpubs.htm {\unhbox0
• R Charney, J Crisp, K Vogtmann, Automorphisms of $2$–dimensional right-angled Artin groups, Geom. Topol. 11 (2007) 2227–2264
• M Culler, K Vogtmann, Moduli of graphs and automorphisms of free groups, Invent. Math. 84 (1986) 91–119
• M B Day, Symplectic structures on right-angled Artin groups: Between the mapping class group and the symplectic group, Geom. Topol. 13 (2009) 857–899
• P J Higgins, R C Lyndon, Equivalence of elements under automorphisms of a free group, J. London Math. Soc. $(2)$ 8 (1974) 254–258
• 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
• J McCool, A presentation for the automorphism group of a free group of finite rank, J. London Math. Soc. $(2)$ 8 (1974) 259–266
• J McCool, Some finitely presented subgroups of the automorphism group of a free group, J. Algebra 35 (1975) 205–213
• J Milnor, Introduction to algebraic $K$–theory, Annals of Math. Studies 72, Princeton University Press (1971)
• E S Rapaport, On free groups and their automorphisms, Acta Math. 99 (1958) 139–163
• J-P Serre, Trees, Springer, Berlin (1980) Translated from the French by J Stillwell
• H Servatius, Automorphisms of graph groups, J. Algebra 126 (1989) 34–60
• J H C Whitehead, On equivalent sets of elements in a free group, Ann. of Math. $(2)$ 37 (1936) 782–800