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Splittings of finitely generated groups over two-ended subgroups


Author: Brian H. Bowditch
Journal: Trans. Amer. Math. Soc. 354 (2002), 1049-1078
MSC (2000): Primary 20F65, 20E08
DOI: https://doi.org/10.1090/S0002-9947-01-02907-5
Published electronically: October 26, 2001
MathSciNet review: 1867372
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Abstract: We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of the JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated group which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group.


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Additional Information

Brian H. Bowditch
Affiliation: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton SO17 1BJ, Great Britain

DOI: https://doi.org/10.1090/S0002-9947-01-02907-5
Received by editor(s): January 31, 2001
Received by editor(s) in revised form: July 1, 2001
Published electronically: October 26, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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