Clay, Matt; Conant, John; Ramasubramanian, Nivetha Whitehead graphs and separability in rank two. (English) Zbl 1305.20033 Involve 7, No. 4, 431-452 (2014). 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 PDF BibTeX XML Cite \textit{M. Clay} et al., Involve 7, No. 4, 431--452 (2014; Zbl 1305.20033) Full Text: DOI OpenURL References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.