The deformation space of a simplicial –tree is the set of –trees which can be obtained from by some collapse and expansion moves, or equivalently, which have the same elliptic subgroups as . We give a short proof of a rigidity result by Forester which gives a sufficient condition for a deformation space to contain an –invariant –tree. This gives a sufficient condition for a JSJ splitting to be invariant under automorphisms of . More precisely, the theorem claims that a deformation space contains at most one strongly slide-free –tree, where strongly slide-free means the following: whenever two edges incident on a same vertex are such that , then and are in the same orbit under .
"A very short proof of Forester's rigidity result." Geom. Topol. 7 (1) 321 - 328, 2003. https://doi.org/10.2140/gt.2003.7.321