Finite graphs of groups with isomorphic fundamental groups.

*(English. Russian original)*Zbl 0793.20017
Algebra Logic 30, No. 5, 389-409 (1991); translation from Algebra Logika 30, No. 5, 595-623 (1991).

The fundamental group of a graph of groups provides a presentation for a group acting on a tree (in terms of amalgamated products and HNN- extensions of the vertex and edge stabilizers). Conversely it allows to recover from given “quotient data” (i.e. a graph of groups) a tree and the action of the group on it.

In the present paper reduced graphs of groups satisfying the following restrictions are considered: (i) the graph is finite, (ii) all vertex groups are \((FA)\)-groups (i.e. cannot act on a tree without a fixed point of the whole group) (iii) all edge groups are cohopfian (i.e. not isomorphic to a proper subgroup). The author shows that two such graphs of groups have isomorphic fundamental groups if and only if one can be obtained from the other by a finite number of “elementary transformations” of the following two types: (a) sliding an edge along a second edge, (b) sliding an edge along the vertex group of its end point w.r.t. an element of this vertex group.

Under the same assumptions the reviewer [Arch. Math. 51, No. 3, 232-237 (1988; Zbl 0656.20038)] had claimed a characterization involving only transformations of type (a). The author shows by an example that this is not sufficient. He replaces the reviewer’s faulty “bijection between reduced paths” by an isomorphism between the “dual graphs” (whose vertices are the conjugacy classes of edge groups, and whose edges are given by inclusion).

It should be mentioned that the English translation contains lots of mistakes of all kinds and can only be understood by consulting the original Russian text.

In the present paper reduced graphs of groups satisfying the following restrictions are considered: (i) the graph is finite, (ii) all vertex groups are \((FA)\)-groups (i.e. cannot act on a tree without a fixed point of the whole group) (iii) all edge groups are cohopfian (i.e. not isomorphic to a proper subgroup). The author shows that two such graphs of groups have isomorphic fundamental groups if and only if one can be obtained from the other by a finite number of “elementary transformations” of the following two types: (a) sliding an edge along a second edge, (b) sliding an edge along the vertex group of its end point w.r.t. an element of this vertex group.

Under the same assumptions the reviewer [Arch. Math. 51, No. 3, 232-237 (1988; Zbl 0656.20038)] had claimed a characterization involving only transformations of type (a). The author shows by an example that this is not sufficient. He replaces the reviewer’s faulty “bijection between reduced paths” by an isomorphism between the “dual graphs” (whose vertices are the conjugacy classes of edge groups, and whose edges are given by inclusion).

It should be mentioned that the English translation contains lots of mistakes of all kinds and can only be understood by consulting the original Russian text.

Reviewer: F.Herrlich (Karlsruhe)

##### MSC:

20E08 | Groups acting on trees |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20F34 | Fundamental groups and their automorphisms (group-theoretic aspects) |

##### Keywords:

cohopfian edge groups; elementary transformations; fundamental group of a graph of groups; tree; amalgamated products; HNN-extensions; reduced graphs of groups; \((FA)\)-groups
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\textit{D. G. Khramtsov}, Algebra Logic 30, No. 5, 389--409 (1991; Zbl 0793.20017); translation from Algebra Logika 30, No. 5, 595--623 (1991)

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##### References:

[1] | W. Magnus, A. Karrass, and D. Solitar, Combinatorial Group Theory, Interscience, New York (1966). |

[2] | R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Springer-Verlag, Berlin (1977). |

[3] | H. Bass, ”Some remarks on groups, acting on trees,” Commun. Algebra,4, 1091–1126 (1976). · Zbl 0383.20021 |

[4] | F. Herrlich, ”Graphs of groups with isomorphic fundamental groups,” Arch. Math.,51, No. 3, 232–237 (1988). · Zbl 0656.20038 |

[5] | S. Krstič, ”Actions of finite groups on graphs and related automorphisms of free groups,” J. Algebra,124, No. 1, 119–138 (1989). · Zbl 0675.20025 |

[6] | J.-P. Serre, ”Arbres, amalgames,SL 2,” Astérisque,46 (1977). · Zbl 0369.20013 |

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