Walker, John
(2010)
Generalisations of the almost stability theorem.
PhD thesis, University of Glasgow.
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Abstract
This thesis is concerned with the actions of groups on trees and their
corresponding decompositions. In particular, we generalise the Almost Stability Theorem of Dicks and Dunwoody
[12] and an associated application of Kropholler [23] on when a group of finite cohomological
dimension splits over a Poincare duality subgroup.
In Chapter 1 we give a brief overview of this thesis, some historical background information and also
mention some recent developments in this area.
Chapter 2 consists mostly of introductory material, covering group actions on trees,
commensurability of groups and completions of certain spaces. The chapter concludes with a discussion of a
certain completion introduced in [23] and when this has an underlying group structure.
We then introduce the Almost Stability Theorem in Chapter 3 mentioning some possible directions in
which the result may be generalised, how these various conjectures are related and some preliminary results
suggesting that such generalisations are plausible. We go on to state the most general version of the theorem
currently obtained. The proof of this result, Theorem A, takes up the bulk of Chapter 4 which is
based on the approach of the book by Dicks and Dunwoody [12]. In removing the finite edge stabiliser
condition we place certain restrictions on the groups that are allowed.
Finally, in Chapter 5 we investigate Poincare duality groups, the connection between outer
derivations and almost equality classes and show how to use Theorem A to obtain a more general version of the
results of Kropholler. This work culminates in the result that Theorem B is a corollary of Theorem A.
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