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[43] M.J.Dunwoody, Rectangle groups. (2007)
pdf.file.
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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.
[44] M.J.Dunwoody and B.Kroen, Vertex cuts. (2014)
pdf.file.
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Abstract
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.
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[45] M.J.Dunwoody, An ianaccessible graph (2009)
pdf.file.
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Abstract
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.
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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] A.N.Bartholomew and M.J.Dunwoody, Proper decompositions of finitely
presented groups. (2012)
pdf.file.
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Abstract
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.
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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.
[49] M.J.Dunwoody, Structure trees and networks.
(2014)
pdf.file.
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Abstract
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.