Bounding the complexity of simplicial group actions on trees. (English) Zbl 0724.20019

Bestvina, Mladen; Feighn, Mark

Invent. Math. 103, No. 3, 449-469 (1991).

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)].

Reviewer: B.N.Apanasov (Norman)

MSC:

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

Keywords:

simplicial group actions on trees; Grushko’s theorem; accessibility theorem; finitely presented group; reduced G-tree; small edge stabilizers; number of vertices

Citations:

Zbl 0572.20025

References:

[1]	Bass, H.: Some remarks on group actions on trees. Comm. Algebra4, 1091-1126 (1976) · Zbl 0383.20021 · doi:10.1080/00927877608822154
[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 · doi:10.1112/plms/s3-55.3.571
[4]	Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math.81, 449-457 (1985) · Zbl 0572.20025 · doi:10.1007/BF01388581
[5]	Dunwoody, M.J., Fenn, R.A.: On the finiteness of higher knot sums. Topology26, 337-343 (1987) · Zbl 0623.57013 · doi:10.1016/0040-9383(87)90005-X
[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 · doi:10.1007/BF02095993