#### Volume 7, issue 1 (2003)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Ethics Statement Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060 Other MSP Journals
A very short proof of Forester's rigidity result

### Vincent Guirardel

Geometry & Topology 7 (2003) 321–328
 arXiv: math.GR/0301284
##### 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 ${G}_{v}$.

##### Keywords
tree, graph of groups, folding, group of automorphisms
##### Mathematical Subject Classification 2000
Primary: 20E08
Secondary: 57M07, 20F65