On the smallest non-trivial quotients of mapping class groups

Abstract

We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus g \geq 3 without punctures is Sp _{2g}(2) , thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz’s results on \mathbb C -linear representations of mapping class groups to projective representations over any field.

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.…”

“…2 however, these will not be surjective in general, as any matrix in GL(n, Z) has determinant ±1, but there can be more units in Zm (this issue does not arise for the SL(n, Z) setting). 3 which corresponds to all of Z n 2 .…”

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

“…Zimmermann also formulated a corresponding conjecture for mapping class groups; this has now been solved by the second and third authors in .…”

Small Quotients of Braid Groups

“…Zimmermann [10] proved that for g ∈ {3, 4}, the smallest minimal non-trivial quotient of the genus g mapping class group Mod(S g ) is the symplectic group Sp g (Z/2). He then conjectured that the result held for g ≥ 3, which was later proven by Kielak-Pierro [6]. Margalit's question can be seen as the braid group analogue of Zimmermann's.…”