This thesis is concerned with the actions of groups on trees and theircorresponding decompositions. In particular, we generalise the Almost Stability Theorem of Dicks and Dunwoody[12] and an associated application of Kropholler [23] on whena group of finite cohomologicaldimension splits over a Poincare duality subgroup.In Chapter 1 we give a brief overview of this thesis, some historical background information and alsomention 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 acertain 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 inwhich the result may be generalised, how these various conjectures are related and some preliminary resultssuggesting that such generalisations are plausible. We go on to state the most general version of the theoremcurrently obtained. The proof of this result, Theorem A, takes up the bulk of Chapter 4 which isbased on the approach of the book by Dicks and Dunwoody [12]. In removing the finite edge stabilisercondition we place certain restrictions on the groups that are allowed.Finally, in Chapter 5 we investigate Poincare duality groups, the connection between outerderivations and almost equality classes and show how to use Theorem A to obtain a more general version of theresults of Kropholler. This work culminates in the result that Theorem B is a corollary of Theorem A.
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