Illinois Journal of Mathematics

The automorphism group of a graph product with no SIL

Ruth Charney, Kim Ruane, Nathaniel Stambaugh, and Anna Vijayan

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We study the automorphisms of graph products of cyclic groups, a class of groups that includes all right-angled Coxeter and right-angled Artin groups. We show that the group of automorphisms generated by partial conjugations is itself a graph product of cyclic groups providing its defining graph does not contain any separating intersection of links (SIL). In the case that all the cyclic groups are finite, this implies that the automorphism group is virtually $\operatorname{CAT}(0)$; it has a finite index subgroup which acts geometrically on a right-angled building.

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Illinois J. Math., Volume 54, Number 1 (2010), 249-262.

First available in Project Euclid: 9 March 2011

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Zentralblatt MATH identifier

Primary: 20F28: Automorphism groups of groups [See also 20E36] 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 20F36: Braid groups; Artin groups


Charney, Ruth; Ruane, Kim; Stambaugh, Nathaniel; Vijayan, Anna. The automorphism group of a graph product with no SIL. Illinois J. Math. 54 (2010), no. 1, 249--262. doi:10.1215/ijm/1299679748.

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  • A. Barnhill and A. Thomas, Density of commensurators for uniform lattices of right-angled buildings, available at arXiv:0812.2280.
  • K. U. Bux, R. Charney and K. Vogtmann, Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups, Groups, Geometry, and Dynamics 3 (2009), 525–539.
  • R. Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007), 141–158.
  • R. Charney, J. Crisp and K. Vogtmann, Automorphisms of 2-dimensional right-angled Artin groups, Geom. Topol. 11 (2007), 2227–2264.
  • R. Charney and M. Davis, The $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995), 597–627.
  • R. Charney and K. Vogtmann, Finiteness properties of automorphism groups of right-angled Artin groups, Bull. London Math. Soc. 41 (2009), 94–102.
  • L. J. Corredor and M. A. Gutierrez, A generating set for the automorphism group of the graph product of Abelian groups, available at arXiv:0911.0576.
  • M. Davis, The geometry and topology of Coxeter groups, London Math. Soc. Monagr. Ser., vol. 32, Princeton University Press, Princeton, NJ, 2008.
  • M. Davis, Buildings are ${\rm CAT}(0)$, Geometry and Cohomology Group Theory (Durham, 1994), London Math. Soc. Lecture Note Ser., vol. 252, Cambridge University Press, Cambridge, 1998, pp. 108–123.
  • M. Davis, Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. (2) 117 (1983), 293–324.
  • M. Day, Peak reduction and finite presentations for automorphism groups of right-angled Artin groups, Geom. Topol. 13 (2009), 817–855.
  • E. Green, Graph products of groups, Thesis, The University of Leeds, 1990.
  • M. Gutierrez and A. Kaul, Automorphisms of right-angled Coxeter groups, Int. J. Math. Math. Sci. (2008), Art. ID 976390, 10 pp.
  • M. Gutierrez and A. Piggott, Rigidity of graph products of Abelian groups, Bull. Aust. Math. Soc. 77 (2008), 187–196.
  • M. Gutierrez, A. Piggott and K. Ruane, On the automorphisms of a graph product of Abelian groups, to appear in Groups, Geometry, and Dynamics, arXiv:0710.2573.
  • F. Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209.
  • F. Haglund and F. Paulin, Constructions arborescentes d'immeubles, Math. Ann. 325 (2003), 137–164.
  • M. Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. (2) 52 (1995), 318–334.
  • M. Laurence, Automorphisms of graph products of groups, Thesis, QMW College, University of London, 1992.
  • B. Mühlherr, Automorphisms of graph-universal Coxeter groups, J. Algebra 200 (1998), 629–649.
  • H. Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), 34–60.
  • A. Thomas, Lattices acting on right-angled buildings, Algebr. Geom. Topol. 6 (2006), 1215–1238.
  • J. Tits, Sur le groupe des automorphismes de certains groupes de Coxeter, J. Algebra 113 (1988), 346–357.