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Infinite presentability of groups and condensation

Part of: Structure and classification of infinite or finite groups Special aspects of infinite or finite groups Connections with homological algebra and category theory

Published online by Cambridge University Press:  02 January 2014

Robert Bieri ,
Yves Cornulier ,
Luc Guyot  and
Ralph Strebel
Robert Bieri
Affiliation:
Department of Mathematics, Johann Wolfgang Goethe-Universität Frankfurt, 60054 Frankfurt am Main, Germany ([email protected])
Yves Cornulier
Affiliation:
CNRS and Laboratoire de Mathématiques, Bâtiment 425, Université Paris-Sud 11, 91405 Orsay, France ([email protected])
Luc Guyot
Affiliation:
STI, EPFL, INN 238 Station 14, Lausanne 1015, Switzerland ([email protected])
Ralph Strebel
Affiliation:
Département de Mathématiques, Chemin du Musée 23, Université de Fribourg, 1700 Fribourg, Switzerland ([email protected])
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Abstract

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We introduce here a larger class of condensation groups, called infinitely independently presentable groups, and establish criteria which allow one to infer that a group is infinitely independently presentable. In addition, we construct examples of finitely generated groups with no minimal presentation, among them infinitely presented groups with Cantor–Bendixson rank 1, and we prove that every infinitely presented metabelian group is a condensation group.

Keywords

Cantor–Bendixson rankcondensation groupinfinitely presented metabelian groupinvariant SigmaThompson’s group Fspace of marked groups

MSC classification

Secondary: 20F05: Generators, relations, and presentations 20E05: Free nonabelian groups 20E06: Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations 20F16: Solvable groups, supersolvable groups 20J06: Cohomology of groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 13 , Issue 4 , October 2014 , pp. 811 - 848
Copyright
©Cambridge University Press 2013 

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