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.

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Vincent Guirardel
Laboratoire E. Picard, UMR 5580, Batiment 1R2
Universite Paul Sabatier, 118 rte de Narbonne
31062 Toulouse cedex 4, France

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