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Peripheral splittings of groups
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Bowditch, B. H.. (2001) Peripheral splittings of groups. American Mathematical Society. Transactions, Vol.353 (No.10). pp. 40574082. ISSN 00029947
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Official URL: http://dx.doi.org/10.1090/S0002994701028355
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 nontrivial peripheral splitting, then its boundary has a global cut point.
Moreover, the nonperipheral 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:  00029947 
Official Date:  2001 
Volume:  Vol.353 
Number:  No.10 
Page Range:  pp. 40574082 
Identification Number:  10.1090/S0002994701028355 
Status:  Peer Reviewed 
Access rights to Published version:  Open Access 
References:  [AN] S.A. Adeleke, P.M. Neumann, Relations related to betweenness: their structure and automorphisms, 
URI:  http://wrap.warwick.ac.uk/id/eprint/4292 
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