A very short proof of Forester’s rigidity result

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 $G_{e_{1}} \subset G_{e_{2}}$ , 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

References

Forward citations

Publication

Received: 24 January 2003

Revised: 11 April 2003

Accepted: 14 May 2003

Published: 19 May 2003

Proposed: Walter Neumann

Seconded: Cameron Gordon, David Gabai

Authors

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

Volume 7, issue 1 (2003)

Vincent Guirardel