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Let $X=(x_1, \ldots, x_n)$ be an $n$-tuple of elements of a given group $G$. Then two $n$-tuples $X$ and $Y$ are Nielsen equivalent if there exists an automorphism of the free group on $n$-generators, $\phi\in \operatorname{Aut}(F_n)$, such that $X\phi=(x_1\phi, \ldots, x_n\phi)=Y$. Also, $X$ and $Y$ are said to lie in the same $T$-system if there exists an automorphism of $G$, $\psi\in \operatorname{Aut}(G)$, such that $X\psi$ is Nielsen equivalent to $Y$.

It is (relatively) well-known that both "Nielsen equivalence" and "lying in the same $T$-system" are equivalence relations.

I am interested in what is known about generating $n$-tuples of one-relator groups, so groups of the form $\langle x_1, \ldots, x_n; R^m\rangle$ where $m\geq 1$ and $R\neq S^i$ for any words $S\in F(X)$ and $i>1$ (so $R$ is not a proper power of any element of the free group).

Now, I know that if $n=2$ and $m>1$ then there is only one Nielsen equivalence class (this is in a paper from the 70s which uses this fact to solve the isomorphism problem for such groups - "The isomorphism problem for two-generator one-relator groups is solvable"), and I know that there is a recent paper out there somewhere about what happens for $n$ finite and $m>1$ (although I cannot for the life of me find it...I think it is written by Schupp and some others, but I can't remember!). However, I was wondering what happens if $m=1$. My question is the following,

Let $G=\langle X; R^m\rangle$, $|X|=2$, $m=1$. What can be said about the number of Nielsen equivalence classes in the $T$-system of $X$? Are there finitely many, or infinitely many? What if $|X|>2$?

Note that it is well-known that if $m>1$ then these groups are residually finite (whether or not the 185 page proof is this result is true or not is beside the point!) and so generating them should be easy. However, if $m=1$ then you might end up with non-Hopfian groups, never mind non-residually finite! So, the epimorphism which is surjective but not injective in your group will (if I remember correctly) necessarily move your generating tuple out of its $T$-system. Certainly, this happens with the Baumslag-Solitar group $BS(2, 3)$ and the map $a\mapsto a$, $b\mapsto b^2$. Indeed, this map takes your generating tuple to a new $T$-system every time! That is, the generating pairs $(a, b^{2^n})$ lie in different $T$-systems for all $n$. Clearly this means there are infinitely many Nielsen equivalence classes, and so infinitely many $T$-systems. This is why I want to know about what happens in your given $T$-system!

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$X\subset G$ or $X\subset F_n$? –  Boris Novikov Nov 8 '11 at 14:57
2  
Perhaps you already know this, but Gerhard Rosenberger has done a lot of work in this area, so you might find something of use among his papers. –  James Nov 8 '11 at 16:14
    
@Boris Novikov: $X$ is an $n$-tuple of words in the free group $F_n$. –  user6503 Nov 8 '11 at 17:40

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