Transitive actions of $Aut(F_2)$ on surjections from $F_2\twoheadrightarrow G$

Question

Here $F_2$ is the free group on two generators $x,y$. I'm interested in examples of finite groups $G$ such that $Aut(F_2)$ acts transitively on the set of surjections $F_2\rightarrow G$. (In particular I'm interested in $G$'s where automorphisms of $F_2$ of determinant 1 act transitive on surjections $F_2\rightarrow G$)

In this case the conjugacy class of the image of the commutator $xyx^{-1}y^{-1}$ is an invariant of any $Aut(F_2)$-orbit, so a necessary condition for transitivity of the action is that any two commutators of generating pairs of $G$ are conjugate in $G$

If $G$ is simple then one expects that "most" pairs of elements of $G$ will actually generate $G$. Thus, for alternating groups $A_n$ ($n > 5$), where every element is a commutator, one expects the action of $Aut(F_2)$ can't possibly be transitive.

For groups of the form $G = PSL_2(\mathbb{F}_q)$ it's proven in http://www2.math.ou.edu/~dmccullough/research/pdffiles/traces.pdf

that for $q > 11$ almost every element of $\mathbb{F}_q$ appears as the trace of a commutator of a generating pair, and so again $Aut(F_2)$ can't act transitively in this case.

Is this generally true for all finite nonabelian simple groups? More precisely..

1. Does there exist a nonabelian simple group $G$ on which $Aut(F_2)$ acts transitively on surjections $F_2\rightarrow G$?

2. What heuristics are there for describing the groups $G$ admitting such a transitive action (by $Aut(F_2)$? by automorphisms of determinant 1?)

Answer 1

The answer to the first part of OP's question is "no", but possibly with a finite number of exceptions.

Indeed [Theorem 1.8, 4] established that the number of $T_2$-systems of a non-abelian finite simple group $G$ tends to infinity as the cardinality of $G$ tends to infinity. A $T_2$-system in the sense of B. H. Neumann and H. Neumann (T-system stands for transitivity system) is an orbit of $\operatorname{Aut}(F_2) \times \operatorname{Aut}(G)$ acting on the set $\operatorname{Epi}(F_2, G)$ of the epimorphisms $\pi: F_2 \twoheadrightarrow G$ in the following way: $\pi \cdot (\psi, \phi) = \phi^{-1} \circ \pi \circ \psi$. An orbit of $\operatorname{Aut}(F_2)$ is called a Nielsen equivalence class and clearly the number of such classes is bounded from below by the number of $T_2$-systems.

This result was already proved by M. Evans, R. M. Guralnick and I. Pak for the sequence $(PSL_2(\mathbb{F}_q))_q$ with $q$ the power of a prime, and was refined in [5], as noted in OP's question. I. Pak also proved in [Proposition 2.5.2, 1] the result for the sequence $(A_n)_n$ of the finite alternating groups and he conjectured with Guralnick the aforementioned general result in [2]. Estimates for the number of $T_2$-systems of these groups, and for the Suzuki groups as well, are collected in [Theorem 2.1, 3].

I don't know of any non-abelian finite simple group with exactly one $T_2$-system. In contrast, Wiegold's conjecture asserts that the number of $T_3$-systems should always be $1$ for such groups.

Regarding the second part of OP's question, I don't have any heuristics at hand. Stating the obvious, I would definitely try to know more about the classification of finite simple groups.

If the second part of the question concerns also two-generated finite groups which are not necessarily simple, then the following observation may help: the class of the finite groups $G$ for which $\operatorname{Aut}(F_2)$ acts transitively (the same holds if we require the determinant to be $1$) on $\operatorname{Epi}(F_2, G)$ is stable under quotient. This follows from Gaschutz'lemma [Lemma 2.1.5, 1] and imposes for instance that the order of the first invariant factor of $G_{ab} = G/[G, G]$ lies in $\{2, 3, 4, 6\}$ when $G_{ab}$ is not cyclic.

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[1] "What do we know about the product replacement algorithm?", I. Pak, 2000.
[2] "On a question of B.H. Neumann", R. M. Guralnick and I. Pak, 2002.
[3] "Nielsen equivalence and stability graph of finitely generated groups", M. Evans, 2008.
[4] "Commutator maps, measure preservation, and T-system", S. Garion and A. Shalev, 2009.
[5] "Nielsen equivalence of generating pairs of $SL_2(q)$", D. Mc Cullough and M. Wanderley, 2013.

Answer 2

This perhaps should be a comment, but it is a bit too long. In

N. Dunfield, W. Thurston's, Finite covers of random 3-manifolds, Invent. Math. 166 (2006) 457-521

they study the set of surjections from the fundamental group of a genus $g$ surface to a fixed finite group. The automorphism group of a surface group is the mapping class group of a surface. They prove that for large $g$, the number of mapping class group orbits of surjections is independent of $g$ (and in fact give a precise formula for the number of such orbits). See the discussion around Corollary 6.17.

There is a strong analogy between the mapping class group and the automorphism group of a free group, so this discussion might be enlightening.