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**Whitehead graphs and separability in rank two.**
*(English)*
Zbl 1305.20033

Authors’ summary: By applying an algorithm of Stallings regarding separability of elements in a free group, we give an alternative approach to that of R. P. Osborne and H. Zieschang [Invent. Math. 63, 17-24 (1981; Zbl 0438.20017)] in describing all primitive elements in the free group of rank 2. As a result, we give a proof of a classical result of Nielsen, used by Osborne and Zieschang in their work, that the only automorphisms of \(F_2\) that act trivially on the Abelianization are those defined by conjugation. Finally, we compute the probability that a Whitehead graph in rank 2 contains a cut vertex. We show that this probability is approximately \(1/l^2\), where \(l\) is the number of edges in the graph.

Reviewer: Behrooz Mashayekhy (Mashhad)

### MSC:

20E05 | Free nonabelian groups |

20F05 | Generators, relations, and presentations of groups |

20E45 | Conjugacy classes for groups |

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

20E36 | Automorphisms of infinite groups |

20F65 | Geometric group theory |

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

20P05 | Probabilistic methods in group theory |

### Keywords:

free groups; primitive elements; separable elements; Whitehead graphs; cut vertices; algorithms### Citations:

Zbl 0438.20017
Full Text:
DOI

### References:

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