Abstract
Planar locally finite graphs which are almost vertex transitive are discussed. If the graph is 3-connected and has at most one end then the group of automorphisms is a planar discontinuous group and its structure is well-known. A general result is obtained for such graphs where no restriction is put on the number of ends. It is shown that such a graph can be built up from one ended or finite planar graphs in a precise way. The results give a classification of the finitely generated groups with planar Cayley graphs.

[42] M.J.Dunwoody, Isometries and morphisms of real trees. (2007) pdf.file.

Commentary on [38], [39] and [42]
These papers are concerned with the actions of finitely generated and finitely presented groups on R-trees.
In [38] an example is given of a finitely presented group that has an unstable action on an R-tree with infinite cyclic arc stabilizers. It also has incompatible actions on simplicial trees with small edge groups. The group has a genus two orientable surface group P as a homomorphic image, and the above properties are obtained by considering the action of P on the hyperbolic plane.
In [39] a general result is obtained showing that if a finitely presented group has a non-trivial action on an R-tree T with slender arc stabilizers then it has a decomposition as a fundamental group of a graph of groups in which each vertex group V either fixes a point of T, has a normal subgroup N such that V/N is a group of isometries of R or it is a group similar to the one described in [38]. This generalizes the result of Bestvina and Feighn for stable actions of finitely presented groups on R-trees. It is also shown that the above result of Bestvina and Feighn can be extended to stable actions of finitely generated groups on R-trees.
Two generator group actions on R-trees in which each generator induces a hyperbolic isometry are investigated in [42]. It is also required that the axes intersect and the two hyperbolic lengths are independent over the rationals. The axes will intersect in a closed segment which may be of infinite length. Such actions are classified, as are the groups which act in this way. If in addition arc stabilizers are small then the classification shows that there are only a very limited number of groups and actions. This makes if possible to strengthen the result of [39] for a finitely presented group acting on an R-tree with slender arc stabilizers so that it is for a finitely presented group acting on an R-tree with small arc stabilizers.

[43] M.J.Dunwoody, Rectangle groups. (2007) pdf.file.

Abstract
A class of groups is investigated, each of which has a fairly simple presentation . For example the group
R = (a, b, c, d | a^2 = b^2 = c^2 = d^3 = 1, ba^{-1} =dc^{-1}, ca^{-1} = db^{-1} )
is in the class. Such a group does not have any surface group as a homomorphic image. However it does have incompatible splittings over subgroups which are not small. This contradicts some ideas I had about universal JSJ decompostions for finitely presented groups over finitely generated subgroups. Such a group also has an unstable action on an R-tree and a cocompact action on a CAT(0) cube complex with finite cyclic point stabilizers.

[46] M.J.Dunwoody, Finitely presented groups acting on trees. (2012, This version February 2022) pdf.file.

Abstract
It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there is no global fixed point, then $G$ has a non-trivial action on a simplicial $\R $-tree.

[47] M.J.Dunwoody, An (FA)-group that is not (FR). (2013, This version April, 2014) pdf.file.

Abstract
An example is given of a finitely generated group $L$ that has a non-trivial action on an $\R $-tree but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto $L$ does have a fixed point-free action on some simplicial tree. This is a corrected version of the paper that previously appeared on arXiv. coauthored with A. Minasyan.

[50] A.N.Bartholomew and M.J.Dunwoody, Proper decompositions of finitely presented groups. (this version February 2022) pdf.file.

Abstract
This is a report on our long term project to find an algorithm to decide if a finitely presented group has a non-trivial action on a tree.

[51] M.J.Dunwoody, The Kropholler conjecture, (November 2023) pdf.file.

Abstract
The first version of this paper, gave another proof of the Kropholler Conjecture, which gives a relative version of Stallings' Ends Theorem, following an earlier incorrect proof. It has been pointed out by Sam Shepherd that the second proof was still inadequate. We explain the difficulty and possible ways to obtain a correct proof.

[52] M.J.Dunwoody, Ends and accessibility, pdf.file.
Abstract
An account is given of how I became involved with Stallings' Theorem on the ends of groups and related subsequent developments relating to Wall's Conjecture on the accessibility of finitely generated groups.

[53] M.J.Dunwoody, A short proof of the Poincar\'e Conjecture, pdf.file.
Abstract
A short, fairly self-contained proof is given of the Poincar\' e Conjecture. As of 24-03-25 no fatal mistake in this proof has been pointed out to the author.

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