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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. 4057-4082. ISSN 0002-9947

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Official URL: http://dx.doi.org/10.1090/S0002-9947-01-02835-5

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

Official Date:

2001

Dates:

Date	Event
2001	Published

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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