A very short proof of Forester's rigidity result

Abstract

The deformation space of a simplicial $G$–tree $T$ is the set of $G$–trees which can be obtained from $T$ by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as $T$. We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an $Aut\left(G\right)$–invariant $G$–tree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of $G$. More precisely, the theorem claims that a deformation space contains at most one strongly slide-free $G$–tree, where strongly slide-free means the following: whenever two edges $e_1,e_2$ incident on a same vertex $v$ are such that $G_{e_1}\subset G_{e_2}$, then $e_1$ and $e_2$ are in the same orbit under $G_v$.