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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Peripheral splittings of groups

Author: B. H. Bowditch
Journal: Trans. Amer. Math. Soc. 353 (2001), 4057-4082
MSC (2000): Primary 20F67, 20E08
Published electronically: June 1, 2001
MathSciNet review: 1837220
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Abstract: We define the notion of a “peripheral splitting” of a group. This is essentially a representation of the group as the fundamental group of a bipartite graph of groups, where all the vertex groups of one colour are held fixed—the “peripheral subgroups”. We develop the theory of such splittings and prove an accessibility result. The theory mainly applies to relatively hyperbolic groups with connected boundary, where the peripheral subgroups are precisely the maximal parabolic subgroups. We show that if such a group admits a non-trivial peripheral splitting, then its boundary has a global cut point. Moreover, the non-peripheral vertex groups of such a splitting are themselves relatively hyperbolic. These results, together with results from elsewhere, show that under modest constraints on the peripheral subgroups, the boundary of a relatively hyperbolic group is locally connected if it is connected. In retrospect, one further deduces that the set of global cut points in such a boundary has a simplicial treelike structure.

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

Received by editor(s): November 19, 1999
Received by editor(s) in revised form: January 31, 2001
Published electronically: June 1, 2001
Article copyright: © Copyright 2001 American Mathematical Society