zbMATH — the first resource for mathematics

The accessibility of finitely presented groups. (English) Zbl 0572.20025
A finitely generated group G has more than one end, \(e(G>1\), if there exists a free action of G, with finite quotient, on a simplicial complex K with more than one end. Stallings proved that \(e(G)>1\) iff G splits over a finite subgroup C as a nontrivial free product with amalgamation resp. HNN-extension, \(G=A*_ CB\) or \(G=A*_{\{C,t\}}\) (or equivalently, G acts on a tree such that G fixes no vertex and all edge-stabilizers are finite). Now if e(A) or \(e(B)>1\) this splitting process can be iterated. A f.g. group G is called accessible if this splitting process stops after finitely many steps, or equivalently, G is the fundamental group of a (finite) graph of groups in which every edge group is finite and every vertex group has at most one end. For example, by Grushko’s theorem every f.g. torsion-free group is accessible. The conjecture is that every f.g. group G is accessible.
In the present paper it is shown that finitely presented groups are accessible. The proof is geometric-combinatorial, by splitting an action of G on a suitable simplicial complex K with finite quotient along certain 1-dim. finite subcomplexes which generalize simple closed curves on surfaces. The main point is that this splitting process terminates after finitely many steps; this is analogous to an argument of Kneser who showed that in a compact 3-manifold there are only finitely many disjoint non-parallel embedded 2-spheres. As noted in the introduction, similar methods can be applied to prove the equivariant loop and sphere theorems of Meeks-Yau, thus replacing the minimal surface techniques they used by combinatorial ones [see the author’s paper in Bull. Lond. Math. Soc. 17, 437-448 (1985)].
Reviewer: B.Zimmermann

20F65 Geometric group theory
57M20 Two-dimensional complexes (manifolds) (MSC2010)
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
57M05 Fundamental group, presentations, free differential calculus
20F05 Generators, relations, and presentations of groups
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
Full Text: DOI EuDML
[1] Bieri, R., Strebel, R.: Valuations and finitely presented metabelian groups. Proc. Lond. Math. Soc., (3)41, 439-464 (1980) · Zbl 0448.20029 · doi:10.1112/plms/s3-41.3.439
[2] Dicks, W.: Groups, trees and projective modules. Lect. Notes Math. vol. 790. Berlin-Heidelberg-New York: Springer 1980 · Zbl 0427.20016
[3] Dunwoody, M.J.: Accessibility and groups of cohomological dimension one Proc. Lond. Math. Soc., (3)38, 193-215 (1979) · Zbl 0419.20040 · doi:10.1112/plms/s3-38.2.193
[4] Hempel, J.: 3-Manifolds. Ann. Math. Stud. Princeton: Princeton University Press 1978
[5] Linnell P.A.: On accessibility of groups. J. Pure Appl. Algebra30, 39-46 (1983) · Zbl 0545.20020 · doi:10.1016/0022-4049(83)90037-3
[6] Meeks, W., Yau, S.-T.: The equivariant Dehn’s lemma and loop theorem. Comment. Math. Helv.56, 225-239 (1981) · Zbl 0469.57005 · doi:10.1007/BF02566211
[7] Mecks, W., Yau, S.-T.: The classical Plateau problem and the topology of three-dimensional manifolds. Topology21, 409-442 (1982) · Zbl 0489.57002 · doi:10.1016/0040-9383(82)90021-0
[8] Stallings, J.R.: On torsion-free groups with infinitely many ends. Ann. Math.88, 312-334 (1968) · Zbl 0238.20036 · doi:10.2307/1970577
[9] Stallings, J.R.: Group theory and three dimensional manifolds. Yale Math., monographs no. 4. New Haven: Yale University Press 1971 · Zbl 0241.57001
[10] Wall, C.T.C.: Pairs of relative cohomological dimension one. J. Pure Appl. Algebra1 141-154 (1971) · Zbl 0218.18011 · doi:10.1016/0022-4049(71)90014-4
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.