We study a notion of deformation for simplicial trees with group actions
(–trees). Here
is a fixed, arbitrary
group. Two –trees
are related by a deformation if there is a finite sequence of collapse and
expansion moves joining them. We show that this relation on the set of
–trees
has several characterizations, in terms of dynamics, coarse geometry,
and length functions. Next we study the deformation space of a fixed
–tree
. We show
that if is
“strongly slide-free” then it is the unique reduced tree in its deformation
space. These methods allow us to extend the rigidity theorem of Bass and
Lubotzky to trees that are not locally finite. This yields a unique factorization
theorem for certain graphs of groups. We apply the theory to generalized
Baumslag–Solitar groups and show that many have canonical decompositions.
We also prove a quasi-isometric rigidity theorem for strongly slide-free
–trees.
Keywords
$G$–tree, graph of groups, folding, Baumslag–Solitar group,
quasi-isometry
