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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)
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Abstract

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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
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

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[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