Stable equivalence of generating sets of a finitely-generated group?

Question

This question came about when I was naively considering to what extent generating sets for finitely-generated groups are unique.

Let $G$ be a group and let $\phi_1, \phi_2 : F_k \to G$ be two surjective homomorphisms from a free group on $k$ letters $F_k$. Given a natural number $K \geq k$ let the surjective homomorphisms $\Phi_1^K, \Phi_2^K : F_K \to G$ be the obtained by extending the corresponding maps $\phi_1$ and $\phi_2$ to send the $K-k$ other letters in $F_K$ to the identity.

Does there exist some $K$ such that $\Phi_1^K \circ \alpha = \Phi_2^K$ for some automorphism $\alpha$ of $F_K$?

Answer 1

The answer is $K=2k$, which is an exercise. Here is a sketch: Let $f\colon \langle x_i\rangle\to G$ be a surjective homomorphism, and $h\colon \langle x_i\rangle*\langle y\rangle\to G$ the map defined by $h(x_i)=f(x_i)$, $h(y)=e$. Let $g\in G$ be arbitrary, and choose $w\in\langle x_i\rangle$ such that $f(w)=g$. Let $\alpha$ be the automorphism of $\langle x_i\rangle*\langle y\rangle$ which maps $y$ to $yw$ and leaves the other generators alone. Then $h(\alpha(y))=g$. Do this $k$ times, one for each element of your second generating tuple, then reverse the process, swapping the roles of the $x$ and $y$.

$K=2k-1$ doesn't suffice, by Kapovich and Weidmann, who in "Nielsen equivalence in a class of random groups" construct small-cancellation examples.

Answer 2

The keyword here is "Nielsen equivalence". Two size-$n$ generating sets for a group $G$ are called Nielsen equivalent if the associated surjections $F_n \rightarrow G$ differ by an automorphism of $F_n$. This is an interesting and fairly subtle condition, and it is definitely not completely understood. For instance, there is a famous conjecture of Wiegold saying that if $G$ is a finite simple group, then for $n \geq 3$ any two $n$-element generating sets for $G$ are Nielsen equivalent.

Of course, you are interested in the "stable" version of this where you allow the addition of trivial generators. One theme here is that for many classes of groups, there are many Nielsen equivalence classes of "minimal" generating sets, but only one for non-minimal generating sets (for instance, the ones you get by adding a trivial generator to a minimal generating set).

I hope that other people can give you precise references to the literature on this -- I used to have a decent bibliography of this in my notes from an old collaboration, but I can't find it right now -- but at the very least you can find a lot by searching mathscinet now that you know the right technical term.

(This started off life as a comment, but grew a little too long. Hopefully other people can point you to some papers on this)