Dendrology of groups: An introduction.

*(English)*Zbl 0649.20033
Essays in group theory, Publ., Math. Sci. Res. Inst. 8, 265-319 (1987).

[For the entire collection see Zbl 0626.00014.]

This expository paper is an informal but nonetheless valuable introduction to some aspects of group actions on trees. A large amount of material, due to many authors, is covered and only a brief outline can be given here.

The work is in six sections. Sections 1 and 2 are introductory: the first deals, from a topological point of view, with the Bass-Serre theory of group actions on ordinary (simplicial) trees, the second with its extension to generalized trees or \({\mathbb{R}}\)-trees arising from work of Lyndon, Chiswell and Tits. Section 3 formulates some questions on “which groups act and how”, and from then on the material becomes more specialized. Section 4 covers work of Morgan and the author on connections with hyperbolic geometry and ideas of Thurston. Sections 5 and 6 discuss partial solutions to the questions of section 3, the first concentrating on 3-manifold theory after Stallings, and the second on the rank-two case.

This expository paper is an informal but nonetheless valuable introduction to some aspects of group actions on trees. A large amount of material, due to many authors, is covered and only a brief outline can be given here.

The work is in six sections. Sections 1 and 2 are introductory: the first deals, from a topological point of view, with the Bass-Serre theory of group actions on ordinary (simplicial) trees, the second with its extension to generalized trees or \({\mathbb{R}}\)-trees arising from work of Lyndon, Chiswell and Tits. Section 3 formulates some questions on “which groups act and how”, and from then on the material becomes more specialized. Section 4 covers work of Morgan and the author on connections with hyperbolic geometry and ideas of Thurston. Sections 5 and 6 discuss partial solutions to the questions of section 3, the first concentrating on 3-manifold theory after Stallings, and the second on the rank-two case.

Reviewer: D.J.McCaughan

##### MSC:

20F65 | Geometric group theory |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

57M15 | Relations of low-dimensional topology with graph theory |

05C05 | Trees |

05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |

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