Martin John DUNWOODY Martin John DUNWOODY

Recent papers available electronically

[38] M.J.Dunwoody, A tessellation and a group acting on trees. (2003) dvi.file.

[39] M.J.Dunwoody, Groups acting on real trees. (Version August 2007) pdf.file.

[40] W.Dicks and M.J.Dunwoody, Retracts of vertex sets of trees and the almost stability theorem. J. Group Theory 10 (2007), 703-721. W.Dicks Home Page
Let G be a group, let T be an (oriented) G-tree with finite edge stabilizers, and let VT denote the vertex set of T. We show that, for each G-retract V' of the G-set VT, there exists a G-tree whose edge stabilizers are finite and whose vertex set is V'. This fact leads to various new consequences of the almost stability theorem. We also give an example of a group G, a G-tree T and a G-retract V' of VT such that no G-tree has vertex set V'.

[41] M.J.Dunwoody, Planar graphs and covers. (2007) pdf.file.
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.
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.

[44] M.J.Dunwoody and B.Kroen, Vertex cuts. (2014) pdf.file.

Let X be a connected simple graph. A vertex cut is a connected subset A of VX for which the set NA of vertices which are adjacent to a vertex of A but which are not in A is finite. A ray is a sequence v_1, v_2, , ... of distinct vertices for which v_i is adjacent to v_{i+1} for each i. Two rays are separated by a cut A if one ray is eventually in A and the other is eventually in its complement. Two rays belong to the same end (or vertex end) if they are not separated by any cut. A minimal cut is one which separates rays and for which |NA| is smallest with this property. It is shown that if G is the automorphism group of X then there is a G-tree (called a structure tree) which gives complete information as to the ends of X which are separated by minimal cuts. This theory gives a generalization of Stallings Theorem for the structure of finitely generated groups with infinitely many ends. There is a similar theory for finite graphs in which case the structure tree provides information as to how the graph can be disconnected by removing a set of vertices leaving at least two components which are large (in a certain sense) compared to the disconnecting set. This is a new version of the paper first appearing on this page in 2008.

[45] M.J.Dunwoody, An ianaccessible graph (2009) pdf.file.
An example is given of a locally finite vertex transitive inaccessible graph that is not quasi-isometric to a Cayley graph.

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

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] A.N.Bartholomew and M.J.Dunwoody, Proper decompositions of finitely presented groups. (2012) pdf.file.

A decomposition or splitting of a group $G$ is a decomposition as a free product with amalgamation $G = A*_CB$ for some subgroup $C$ of both $A$ and $B$, or as an HNN-group $G = A*C$. The decomposition is trivial if it is as a free product with amalgamation and $A = C$ or $B= C$.
Sageev has shown how to construct a $G$-cubing associated with a finite number of decompositions of $G$.
The following theorem is proved.
A finitely presented group $G$ has a finite list of $n$ splittings for which the associated $G$-cubing $\tilde C$ has edge and vertex groups which are $G$-unsplittable. Every $G$-unsplittable subgroup of $G$ fixes a vertex of $\tilde C$. The group $G$ has a non-trivial action on a tree if and only if at least one splitting in the list is non-trivial. The list of splittings is computable.

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

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.

[49] M.J.Dunwoody, Structure trees and networks. (2014) pdf.file.

In this paper it is shown that for any network there is a uniquely determined network based on a structure tree that provides a convenient way of determining a minimal cut separating a pair $s, t$ where each of $s, t$ is either a vertex or an end in the original network. A Max-Flow Min-Cut Theorem is proved for any network. In the case of a Cayley Graph for a finitely generated group the theory provides another proof of Stallings' Theorem on the structure of groups with more than one end.

[50] M.J.Dunwoody, Almost Invariant Sets. (2014) pdf.file.

A short proof of a conjecture of Kropholler is given. This gives a relative version of Stallings' Theorem on the structure of groups with more than one end. A generalisation of the Almost Stability Theorem is also obtained, that gives information about the structure of the Sageev cubing.

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Last updated January, 2015.