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A remark on Whitehead’s cut-vertex lemma. (English) Zbl 1441.20016
The review is based on the introduction to this article. J. H. C. Whitehead’s cut-vertex lemma [Proc. Lond. Math. Soc. (2) 41, 48–56 (1936; Zbl 0013.24801; JFM 62.0079.04)] implies that the Whitehead graph of a primitive element of the free group is either not connected or has a cut vertex. Here, an element of a free group is called primitive if it is a member of some basis. This result was generalized by R. Stong [Math. Res. Lett. 4, No. 2–3, 201–210 (1997; Zbl 0983.20023)] and J. R. Stallings [in: Geometric group theory down under. Proceedings of a special year in geometric group theory, Canberra, Australia, July 14–19, 1996. Berlin: de Gruyter 317–330 (1999; Zbl 1127.57300)] to separable subsets of free groups. A set \(S\) of a free group \(F\) is called separable if there exists a non-trivial free factorization \(F=F_1*F_2\) such that each element of \(S\) is conjugate to an element of \(F_1\) or of \(F_2\). In the paper under review, the authors observe that this is an essentially trivial consequence of Stallings folds. To do so the authors relate the connectivity properties of the Whitehead graph of a set of conjugacy classes of elements to the readability of cyclically reduced representatives of these classes in a class of labeled graphs that they call almost-roses.
MSC:
20E05 Free nonabelian groups
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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[1] W. Dicks, On Whitehead’s first free-group algorithm, cutvertices, and free-product factorizations, preprint (2017), .
[2] J. R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551-565. · Zbl 0521.20013
[3] J. R. Stallings, Whitehead graphs on handlebodies, Geometric Group Theory Down Under (Canberra 1996), De Gruyter, Berlin (1999), 317-330. · Zbl 1127.57300
[4] R. Stong, Diskbusting elements of the free group, Math. Res. Lett. 4 (1997), 201-210. · Zbl 0983.20023
[5] J. H. C. Whitehead, On certain sets of elements in a free group, Proc. Lond. Math. Soc. 41 (1936), 48-56. · JFM 62.0079.04
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