#### Geometry & Topology, Vol. 7 (2003)
Paper no. 10, pages 321--328.

## A very short proof of Forester's rigidity result

### Vincent Guirardel

**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(G)-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} is a
subset of 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

**AMS subject classification**.
Primary: 20E08.
Secondary: 57M07, 20F65.

**DOI:** 10.2140/gt.2003.7.321

**E-print:** `arXiv:math.GR/0301284`

Submitted to GT on 24 January 2003.
(Revised 11 April 2003.)
Paper accepted 14 May 2003.
Paper published 19 May 2003.

Notes on file formats
Vincent Guirardel

Laboratoire E. Picard, UMR 5580, Batiment 1R2

Universite Paul Sabatier, 118 rte de Narbonne

31062 Toulouse cedex 4, France

Email: guirardel@picard.ups-tlse.fr

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