Bowditch, B. H. (Brian Hayward), 1961- . (2001) Peripheral splittings of groups. American Mathematical Society. Transactions, Vol.353 (No.10). pp. 4057-4082. ISSN 0002-9947

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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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Hyperbolic groups, Maximal subgroups
Journal or Publication Title:	American Mathematical Society. Transactions
Publisher:	American Mathematical Society
ISSN:	0002-9947
Date:	2001
Volume:	Vol.353
Number:	No.10
Page Range:	pp. 4057-4082
Identification Number:	10.1090/S0002-9947-01-02835-5
Status:	Peer Reviewed
Access rights to Published version:	Open Access
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URI:	http://wrap.warwick.ac.uk/id/eprint/4292

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