Isometric actions on R-trees via pretrees
Shane O'Rourke, Cork Institute of Technology
Saturday May 20, 12.15-1.00, Groups in Galway 2006
Λ-trees (where Λ is a linearly ordered abelian group)
are a generalisation of ordinary trees, where one has the notion of
the distance between two points; in the familiar case
Λ=Z$, the distance function is the path
metric. Pretrees (also known as B-sets) are a further generalisation
of Λ-trees where one has only the notion of a point lying
between two others, and there is a theory of non-nesting actions on
pretrees similar to that of isometric actions on Λ-trees.
It has been shown that under certain conditions -- notably the
archimedean property -- a group that admits an action on a pretree
also admits an isometric action on an R-tree. We show that
given an archimedean action on a pretree X, there is an equivariant
embedding of X in a Λ-tree on which the group acts
isometrically. It follows that if the group is finitely generated and
the action on the pretree is suitably `non-trivial' then there is a
non-trivial (meaning that there is no global fixed point) isometric
action of the group on an R-tree.