# Do the automorphism groups of $F_2$ and $G$ act transitively on the surjections from $F_2$ to $G$?

Let $G$ be a finite nonabelian group.

Let $F_2$ be the free group with generators $x,y$, then we know its outer automorphism group is isomorphic to $\text{GL}_2(\mathbb{Z}$). Let $\text{Aut}^+(F_2)$ be the (index 2) subgroup of $\text{Aut}(F_2)$ mapping onto $\text{SL}_2(\mathbb{Z})\subset\text{GL}_2(\mathbb{Z})$ (via some fixed isomorphism $\text{Out}(F_2)\cong\text{GL}_2(\mathbb{Z})$).

Let's say that $G$ is "nice" if there is a surjection $\rho : F_2\twoheadrightarrow G$ with the property that every surjection $F_2\twoheadrightarrow G$ can be written as $\alpha\circ\rho\circ\varphi$ where $\alpha\in\text{Aut}(G)$, and $\varphi\in\text{Aut}^+(F_2)$.

Say it's "almost nice" if we allow $\varphi\in\text{Aut}(F_2)$.

Is every 2-generated group "nice" or "almost nice"?

Is there some similar class of groups that people have studied? It's important for my research to have a nice characterization of these groups, however I'm not a group theorist and hopefully I won't have to start from scratch when analyzing these groups.

thanks

• Thurston and Dunfield (Inventiones, 2006) looked at the case of surface groups and epimomorphisms to simple finite groups. They proved that (if genus is high w/r to the fixed G) equivalence classes of epimorphisms are determined by a single invariant, the "Euler class". Mar 21, 2014 at 16:32
• Well, $G$ has to be $2$-generated, so lots of $G$ are not "nice" Apr 4, 2014 at 20:36
• thanks, though of course the real question is whether every 2-generated group is "nice/almost nice". Apr 5, 2014 at 0:34
• Some terminology: the words you're looking for are 'Nielsen equivalence'. A group is '(almost?) nice' in your definition if all generating pairs are Nielsen equivalent.
– HJRW
Apr 5, 2014 at 5:16
• Here's a useful fact which will answer your question. (I was just reminded of it in a paper of Kapovich and Weidmann). $\mathrm{Aut}(F_2)$ preserves the conjugacy class of the subgroup generated by the commutator of the generators (since every automorphism can be realized on a punctured torus). So any two generating pairs whose commutators are not conjugate (or conjugate to each other's inverses) provide an example of non-Nielsen-equivalent generating pairs.
– HJRW
Apr 5, 2014 at 5:25

In my answer to your other question, I give references supporting the fact that almost all finite non-abelian simple groups aren't "almost nice".

So I would rather ask: Is there an handy characterization of "(almost) nice" groups?

Since counting the $$T_2$$-systems of finite two-generated nilpotent groups of class $$2$$ is, I believe, still an open problem [Open Problem 2.3.5, 2] and knowing M. J. Dunwoody's result which states that arbitrarily large numbers of $$T_2$$-systems can be achieved in this class [1], I am tempted to say that the task of characterizing "(almost) nice" groups is not obvious.

In this preprint it is shown for instance that split metacyclic groups (e.g., dihedral groups) and wreath products of two finite cyclic groups are "almost nice" (i.e, have only one $$T_2$$-system). The most general result states, when specialized to finite groups, that a two-generated finite split abelian-by-cyclic group is "almost nice", see assertion $$(i)$$ of Theorem $$D$$. The proof is elementary in this case. Since for such groups there is a generating pair $$(a, b)$$ such that $$(a, b) \mapsto (a, b^{-1})$$ induces an automorphism, these groups are actually "nice" groups.

M. J. Dunwoody's construction [1] is the quotient of a split abelian-by-cyclic group by some central subgroup. Hence the class of "almost nice" groups is not stable under quotient and the number of $$T_2$$-systems can dramatically increase under this operation.

[1] "On $$T$$-systems of groups", M. J. Dunwoody, 1962.
[2] "What do we know about the product replacement algorithm?", I. Pak, 2000.