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.


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


Zbl 0438.20017
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