Volume 7, issue 1 (2003)

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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(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 e1,e2 incident on a same vertex v are such that Ge1 Ge2, then e1 and e2 are in the same orbit under Gv.

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