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A very short proof of Forester’s rigidity result

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.
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Contributor : Vincent Guirardel <>
Submitted on : Friday, January 9, 2015 - 3:17:55 PM
Last modification on : Friday, July 10, 2020 - 4:05:11 PM

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Vincent Guirardel. A very short proof of Forester’s rigidity result. Geometry & Topology Monographs, Geometry & Topology Publications, 2003, 7 (1), pp.321-328. ⟨10.2140/gt.2003.7.321⟩. ⟨hal-01101754⟩



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