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If $\varphi$ is an automorphism of $G = \langle x_1, \ldots, x_n; \mathbf{r}\rangle$ such that there exists an automorphism of $F(x_1, \ldots, x_n)$, $\overline{\varphi}$, with $$x_i\varphi=_G x_i\overline{\varphi}$$ for all $1\leq i\leq n$ then $\varphi$ is said to be a free automorphism on $x_1, \ldots, x_n$.

In Magnus, Karrass and Solitar's book Combinatorial Group Theory a whole 3 pages are devoted to the topic of "Free Automorphisms and Free Isomorphisms" (sec. 3.6). However, this section says little more than "they exist", and interestingly that every automorphism can be viewed as one, but not necessarily on the given set of generators (but necessarily on a set of generators no more than twice the size of the given set).

I was therefore wondering if there exists any further results about these automorphisms?

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Another topic to search for is "Nielsen equivalence". – Ian Agol Sep 8 '11 at 20:19
up vote 6 down vote accepted

Such automorphisms are sometimes called "tame". I would recommend asking Vladimir Shpilrain whose email address can be easily found on the Internet (also see his and Gupta's survey Gupta, C. K., Shpilrain, V. Lifting automorphisms: a survey. Groups '93 Galway/St. Andrews, Vol. 1 (Galway, 1993), 249–263, London Math. Soc. Lecture Note Ser., 211, Cambridge Univ. Press, Cambridge, 1995.). There are several recent papers on automorphisms and rigidity of 1-related groups. See, for example, Kapovich, Ilya; Schupp, Paul; Shpilrain, Vladimir Generic properties of Whitehead's algorithm and isomorphism rigidity of random one-relator groups. Pacific J. Math. 223 (2006), no. 1, 113–140.

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Thank you very much. – ADL Sep 13 '11 at 14:36

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