Hostname: page-component-cb9f654ff-kl2l2 Total loading time: 0 Render date: 2025-08-22T23:08:39.937Z Has data issue: false hasContentIssue false
  • English
  • Français

RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

Part of: Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  01 April 2008

MAURICIO GUTIERREZ  and
ADAM PIGGOTT
MAURICIO GUTIERREZ
Affiliation:
Department of Mathematics, Tufts University, 503 Boston Ave, Medford, MA 02155, USA (email: Mauricio.Gutierrez@tufts.edu)
ADAM PIGGOTT
Affiliation:
Department of Mathematics, Tufts University, 503 Boston Ave, Medford, MA 02155, USA (email: Adam.Piggott@tufts.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.

Keywords

graph products of groups

MSC classification

Secondary: 20E34: General structure theorems 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 187 - 196
Copyright
Copyright © 2008 Australian Mathematical Society

References

[1]Droms, C., ‘Isomorphisms of graph groups’, Proc. Amer. Math. Soc. 100(3) (1987), 407–408.CrossRefGoogle Scholar
[2]Green, E. R., ‘Graph products of groups’, PhD Thesis, The University of Leeds, 1990.Google Scholar
[3]Hillar, C. J. and Rhea, D. L., ‘Automorphisms of finite abelian groups’, Amer. Math. Monthly 114(10) (2007), 917–923.CrossRefGoogle Scholar
[4]Laurence, M. R., ‘Automorphisms of graph products of groups’, PhD Thesis, Queen Mary College, University of London, 1993.Google Scholar
[5]Laurence, M. R., ‘A generating set for the automorphism group of a graph group’, J. London Math. Soc. (2) 52(2) (1995), 318–334.CrossRefGoogle Scholar
[6]Radcliffe, D. G., ‘Rigidity of graph products of groups’, Algebr. Geom. Topol. 3 (2003), 1079–1088 (electronic).CrossRefGoogle Scholar