Algebraic & Geometric Topology

Full-featured peak reduction in right-angled Artin groups

Matthew B Day

Full-text: Open access


We prove a new version of the classical peak reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group AΓ on the set of k–tuples of conjugacy classes from AΓ: orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building presentations of stabilizers. This improves on a previous result of the author. We overcome a technical difficulty from the previous work by considering infinite generating sets for the automorphism groups. The method also involves a variation on the Hermite normal form for matrices.

Article information

Algebr. Geom. Topol., Volume 14, Number 3 (2014), 1677-1743.

Received: 5 April 2013
Revised: 19 November 2013
Accepted: 20 November 2013
First available in Project Euclid: 19 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F36: Braid groups; Artin groups
Secondary: 20F28: Automorphism groups of groups [See also 20E36] 15A36

Whitehead algorithm peak reduction automorphism groups of groups right-angled Artin groups raags Hermite normal form


Day, Matthew B. Full-featured peak reduction in right-angled Artin groups. Algebr. Geom. Topol. 14 (2014), no. 3, 1677--1743. doi:10.2140/agt.2014.14.1677.

Export citation


  • M Casals-Ruiz, I Kazachkov, On systems of equations over free partially commutative groups, Mem. Amer. Math. Soc. 212 (2011)
  • R Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007) 141–158
  • R Charney, N Stambaugh, K Vogtmann, Outer space for right-angled Artin groups I
  • H Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics 138, Springer, Berlin (1993)
  • D J Collins, H Zieschang, On the Whitehead method in free products, from: “Contributions to group theory”, (K I Appel, J G Ratcliffe, P E Schupp, editors), Contemp. Math. 33, Amer. Math. Soc. (1984) 141–158
  • D J Collins, H Zieschang, Rescuing the Whitehead method for free products, I: Peak reduction, Math. Z. 185 (1984) 487–504
  • D J Collins, H Zieschang, Rescuing the Whitehead method for free products, II: The algorithm, Math. Z. 186 (1984) 335–361
  • D J Collins, H Zieschang, A presentation for the stabiliser of an element in a free product, J. Algebra 106 (1987) 53–77
  • M B Day, Peak reduction and finite presentations for automorphism groups of right-angled Artin groups, Geom. Topol. 13 (2009) 817–855
  • 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
  • S Hermiller, J Meier, Algorithms and geometry for graph products of groups, Journal of Algebra 171 (1995) 230–257
  • M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. 52 (1995) 318–334
  • H-N Liu, C Wrathall, K Zeger, Efficient solution of some problems in free partially commutative monoids, Inform. and Comput. 89 (1990) 180–198
  • J McCool, A presentation for the automorphism group of a free group of finite rank, J. London Math. Soc. 8 (1974) 259–266
  • J McCool, On Nielsen's presentation of the automorphism group of a free group, J. London Math. Soc. 10 (1975) 265–270
  • J McCool, Some finitely presented subgroups of the automorphism group of a free group, Journal of Algebra 35 (1975) 205–213
  • J Milnor, Introduction to algebraic $K\!$–theory, Annals of Mathematics Studies 72, Princeton University Press (1971)
  • J-P Serre, Trees, 2nd edition, Springer, Berlin (2003)
  • H Servatius, Automorphisms of graph groups, Journal of Algebra 126 (1989) 34–60
  • L VanWyk, Graph groups are biautomatic, J. Pure Appl. Algebra 94 (1994) 341–352
  • J H C Whitehead, On equivalent sets of elements in a free group, Ann. of Math. 37 (1936) 782–800
  • C Wrathall, Free partially commutative groups, from: “Combinatorics, computing and complexity”, (D Z Du, G D Hu, editors), Math. Appl. (Chinese Ser.) 1, Kluwer Acad. Publ., Dordrecht (1989) 195–216