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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.
[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] 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.