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In the book Lyndon, Schupp, Combinatorial Group Theory, P.30 in the edition from 2000 They mention an unpublished work by Waldhausen that is said to give an algorithm to determine whether two subgroups are automorphic given their free generators. I searched for papers written by Waldhausen But I didn't find it. Has Waldhausen or anyone else published a solution to this problem? Where can i find it? This is a copy of a post from StackExchange

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  • $\begingroup$ "give an algorithm to determine whether two subgroups are automorphic given their free generators" is somewhat unclear. Fortunately I have Lyndon-Schupp with me. The general question is: given two $n$-tuples $u$ and $v$ in a f.g. free group $F$ (possibly not free or generating), determine if there exists an automorphism of $F$ mapping $u$ to $v$. Typically this is interesting when $u,v$ are free (and not generating, since this is trivial then). They also mention the question whether there's an automorphism of $F$ mapping $\langle u\rangle$ to $\langle v\rangle$. $\endgroup$
    – YCor
    Commented Mar 8, 2018 at 13:14
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    $\begingroup$ Second given that you were asked clarification on MathSE about your nonstandard use of "automorphic", you should have made an effort to clarify it before reposting here. $\endgroup$
    – YCor
    Commented Mar 8, 2018 at 13:33

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According to a comment on the math.SE link, the OP wants to understand the question of deciding when two (finitely generated) subgroups of a given free group $F$ are equivalent by an automorphism of $F$.

I believe that this was first solved by Gersten:

S. Gersten, On Whitehead’s algorithm, Bull. Am. Math. Soc. 10 (1984) 281– 284.

Abstract. One can decide effectively when two finitely generated subgroups of a finitely generated free group $F$ are equivalent under an automorphism of $F$. The subgroup of automorphisms of $F$ mapping a given finitely generated subgroup $S$ of $F$ into a conjugate of $S$ is finitely presented.

Gersten states that the question asked by the OP was first raised by Whitehead. He also states that he settles this question of Whitehead, and does not cite any papers of Waldhausen. So I guess the paper of Waldhausen never appeared...

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  • $\begingroup$ What do you mean by OP? $\endgroup$
    – Noam Kolodner
    Commented Mar 8, 2018 at 13:38
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    $\begingroup$ @NoamKolodner the 2000 book is just a reprint from the 1977 edition. By the way, Lyndon died in 1988. $\endgroup$
    – YCor
    Commented Mar 8, 2018 at 13:45
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    $\begingroup$ @NoamKolodner Lyndon and Schupp has not been updated since its original publication. $\endgroup$
    – user1729
    Commented Mar 8, 2018 at 13:45
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    $\begingroup$ (...and contains many, many errors...) $\endgroup$
    – user1729
    Commented Mar 8, 2018 at 13:53
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    $\begingroup$ Isn't this just an announcement? Is there a full proof somewhere? $\endgroup$
    – user35370
    Commented Mar 8, 2018 at 19:04

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