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