A very short proof of Forester’s rigidity result

Guirardel, Vincent¹

¹ Laboratoire E. Picard, UMR 5580, Bâtiment 1R2, Université Paul Sabatier, 118 rte de Narbonne, 31062 Toulouse cedex 4, France

Geometry & topology, Tome 7 (2003) no. 1, pp. 321-328.

Résumé

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 $G_v$.

Détail

DOI : 10.2140/gt.2003.7.321

Keywords: tree, graph of groups, folding, group of automorphisms

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