The accessibility of finitely presented groups.

*(English)*Zbl 0572.20025A 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)].

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

##### MSC:

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 |

##### Keywords:

ends of a group; finitely generated group; free action; free product with amalgamation; HNN-extension; graph of groups; finitely presented groups; simple closed curves on surfaces; equivariant loop and sphere theorems##### References:

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