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3-Gaschütz property for epimorphisms from F3 onto finite simple groups

Establish whether every epimorphism from the rank-3 free group F_3 onto a finite simple group is 3-Gaschütz.

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Background

The Wiegold conjecture concerns the transitivity of the Aut(F_n) action on generating n-tuples of finite simple groups. A stronger version (due to Pak) relates directly to lifting generating tuples.

Via Lemma 1.8, the transitivity of Aut(F_n) on generating n-tuples is equivalent to the statement that every epimorphism F_n → G is n-Gaschütz. For n=3 and finite simple groups G, this becomes a central unresolved problem highlighted by the authors.

References

Therefore, while it is known that F i3 not Gaschu ¨tz, the question whether every epi- morphism from F on3o a finite simple group is 3-Gaschu ¨tz is a major open problem.

— Lifting Generators in Connected Lie Groups (2411.12445 - Cohen et al., 19 Nov 2024) in Section 1 (Introduction), The Wiegold Conjecture discussion