Let G be a finite simple group and n ≥ 3. Is the action of Aut(F_n) on Epi(F_n, G) transitive? Research on this question dates back to 1977, and a positive answer is related to the so-called Wiegold conjecture.
For a prime number p, a positive answer is known for the following groups:
PSL(2, p),
Sz(2^{2m−1}) and PSL(2, 2^m), for m ≥ 2,
PSL(2, 3^p),
PSL(2, p^r), for r in N and n ≥ 4,
Alt(k), for k ≤ 11 and n = 3.
We showed that it can also be answered positively for the small Ree groups ^2G_2 (3^{2e+1}), for all e ≥ 1 and all n ≥ 5. This was joint work with Mark Pengitore, Jeroen Schillewaert and Hendrik Van Maldeghem.
In this talk, we provide some insight into the conjecture, discuss some of the methods used to prove our result for the small Ree groups, and explain why it is not easy to resolve all cases.