On the smallest non‐abelian quotient of Aut(Fn)

Abstract

We show that the smallest non‐abelian quotient of Aut(Fn) is prefixPSLnfalse(double-struckZ/2double-struckZfalse)=prefixLnfalse(2false), thus confirming a conjecture of Mecchia–Zimmermann. In the course of the proof we give an exponential (in n) lower bound for the cardinality of a set on which SAut(Fn), the unique index 2 subgroup of Aut(Fn), can act non‐trivially. We also offer new results on the representation theory of SAut(Fn) in small dimensions over small, positive characteristics and on rigidity of maps from SAut(Fn) to finite groups of Lie type and algebraic groups in characteristic 2.

Smallest non-cyclic quotients of the automorphism group of free groups

“…The quotients of smallest orders in these cases are symmetric groups S n and symplectic groups Sp(2g, Z 2 ), respectively, and we have similar results for the quotient maps [4]. Zimmermann [8] proved the result for mapping class groups in the special cases g ∈ {3, 4}, and conjectured the same result for all higher g. This conjecture was proved by Kielak-Pierro [3], using very similar techniques as that of Baumeister-Kielak-Pierro [1] result mentioned earlier. The author gave an elementary proof of these results for both braid and mapping class groups, using the inductive orbit-stabilizer method; and this paper is an analogue for the setting of automorphism group of free groups.…”

“…Mecchia-Zimmermann [7] proved that for n ∈ {3, 4}, the the smallest non-trivial (respectively non-abelian) quotient of SOut(F n ) (respectively Out(F n )) is SL(n, Z 2 ), and conjectured the same statement holds for arbitrary n ≥ 3. This conjecture was proved by Baumeister-Kielak-Pierro [1], using the classification of finite simple groups and representation theory of SAut(F n ). In fact they proved a stronger result for SAut(F n ) and Aut(F n ).…”

On the Smallest Non-trivial Action of SAut(Fn) for Small n

“…Proof. Crucial observations from [BKP,Lemma 2.3], originally due to Bridson-Vogtmann [BV], tell us that all transvections are conjugate in SAut(F n ). In particular, conjugating on the right and given distinct i, j ∈ N we have…”

“…In their paper on non-abelian quotients of Aut(F n ) [BKP,Theorem 3.16], Baumeister, Kielak, and Pierro give a general lower bound for |X| when n 7, as well as specific bounds for n ∈ {3, 4, 5, 6}, with the bounds being sharp when n ∈ {3, 4}. Further, using a result from Saunder's paper on permutation degrees for Coxeter groups [Sau,Theorem 2.3], we can extract a bound of 2n for n 3, which is a greater lower bound for the cases n ∈ {7, 8} than the one given in [BKP,Theorem 3.16].…”

Smallest nontrivial quotients of the commutator subgroup of braid groups

“…The approach of Caplinger-Kordek in [4] is similar to that of Kielak-Pierro [9] and Baumeister-Kielak-Pierro [1] in their proof of analogous conjectures in the setting of mapping class groups and outer automorphism groups of free groups due to Zimmermann [15] and Mecchia-Zimmermann [13], respectively. This approach crucially relies on the classification of finite simple groups, and tries to rule out quotients of smaller order.…”