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On a question of B. H. Neumann

Authors: Robert Guralnick and Igor Pak
Journal: Proc. Amer. Math. Soc. 131 (2003), 2021-2025
MSC (2000): Primary 20D60
Published electronically: December 30, 2002
MathSciNet review: 1963745
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Abstract: The automorphism group of a free group $\mathrm{Aut}(F_k)$ acts on the set of generating $k$-tuples $(g_1,\dots,g_k)$ of a group $G$. Higman showed that when $k=2$, the union of conjugacy classes of the commutators $[g_1,g_2]$ and $[g_2,g_1]$ is an orbit invariant. We give a negative answer to a question of B.H. Neumann, as to whether there is a generalization of Higman's result for $k \ge 3$.

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Additional Information

Robert Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-1113

Igor Pak
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307

Received by editor(s): June 1, 2001
Received by editor(s) in revised form: February 20, 2002
Published electronically: December 30, 2002
Communicated by: Stephen D. Smith
Article copyright: © Copyright 2002 American Mathematical Society