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Bounding the complexity of simplicial group actions on trees. (English) Zbl 0724.20019
The main result of the paper is the following extension of M. J. Dunwoody’s accessibility theorem [see Invent. Math. 81, 449-457 (1985; Zbl 0572.20025)]: Theorem. Let G be a finitely presented group. Then there exists an integer \(\gamma\) (G) such that the following holds: If T is a reduced G-tree with small edge stabilizers, then the number of vertices in T/G is bounded by \(\gamma\) (G). Here a finitely generated group E is small if it does not admit a hyperbolic action on any minimal E-tree. This corresponds to Dunwoody’s condition on edge stabilizers to be either finite or, for a knot group G, infinite cyclic. The theorem fails if G is only required to be finitely generated - see the authors [Proc. Arboreal Group Theory Conf. 1988, Publ., Math. Sci. Res. Inst. 19, 133-141 (1991)].

MSC:
20E08 Groups acting on trees
20F05 Generators, relations, and presentations of groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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References:
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