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)].


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


Zbl 0572.20025
Full Text: DOI EuDML


[1] Bass, H.: Some remarks on group actions on trees. Comm. Algebra4, 1091-1126 (1976) · Zbl 0383.20021
[2] Bestvina, M., Feighn, M.: A counterexample to generalized accessibility. Proceedings of Arboreal Group Theory Conference, MSRI publications (to appear) · Zbl 0826.20027
[3] Culler, M., Morgan, J.: Group actions on ?-trees. Proc. London Math. Soc.55, 571-604 (1987) · Zbl 0658.20021
[4] Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math.81, 449-457 (1985) · Zbl 0572.20025
[5] Dunwoody, M.J., Fenn, R.A.: On the finiteness of higher knot sums. Topology26, 337-343 (1987) · Zbl 0623.57013
[6] Grushko, I.: On the bases of a free product of groups. Mat. Sbornik8, 169-182 (1940)
[7] Scott, G.P., Wall, C.T.C.: Topological methods in group theory. In: Wall, C.T.C. (ed.) Homological Group Theory London Math. Soc. Lect. Notes36, 137-203 (1979) · Zbl 0423.20023
[8] Serre, J.P.: Trees, Springer Berlin Heidelberg New York 1980 · Zbl 0548.20018
[9] Stallings, J.R.: Topology of finite graphs. Invent. Math.71, 551-565 (1983) · Zbl 0521.20013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.