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A very short proof of Forester’s rigidity result. (English) Zbl 1032.20018

Let \(G\) be a group and let \(T\) be a \(G\)-tree. If \(v\) (respectively \(e\)) is a vertex (respectively edge) of \(T\) let \(G_v\) (respectively \(G_e\)) be the stabilizer of \(v\) (respectively of \(e\)). Then \(T\) is strongly slide-free if whenever \(e_1,e_2\) are two edges having a common vertex \(v\) and are such that \(G_{e_1}\subset G_{e_2}\) then \(e_1\) and \(e_2\) are in the same orbit under \(G_v\). The deformation space of \(T\) is the set of \(G\)-trees which can be deformed into \(T\). In this paper there is given a short proof of a theorem of M. Forester [Geom. Topol. 6, 219-267 (2002)] that for each deformation space there is at most one strongly slide-free minimal \(G\)-tree. The spirit of the proof comes from a paper of N. D. Gilbert, J. Howie, V. Metaftsis and E. Raptis [J. Group Theory 3, No. 2, 213-223 (2000; Zbl 0979.20027)].
Reviewer: J.D.Dixon (Ottawa)

MSC:

20E08 Groups acting on trees
57M07 Topological methods in group theory
20F65 Geometric group theory

Citations:

Zbl 0979.20027
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