Splittings of finitely generated groups over two-ended subgroups

Bowditch, Brian

doi:10.1090/S0002-9947-01-02907-5

Splittings of finitely generated groups over two-ended subgroups
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by Brian H. Bowditch

Trans. Amer. Math. Soc. 354 (2002), 1049-1078

DOI: https://doi.org/10.1090/S0002-9947-01-02907-5

Published electronically: October 26, 2001

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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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