Profinite Completions of Burnside-Type Quotients of Surface Groups

Abstract

Using quantum representations of mapping class groups we prove that profinite completions of Burnside-type surface group quotients are not virtually prosolvable, in general. Further, we construct infinitely many finite simple characteristic quotients of surface groups.

Quotients of the mapping class group by power subgroups

“…We first explain how to construct the group from the statement. First, there exists of infinite order in : this is a consequence of work by Koberda–Santharoubane [, Theorem 4.1] for large , and Funar–Lochak [, Proof of Proposition 3.2] for all as in the hypotheses of Theorem . We are going to produce an injective homomorphism as in Lemma , suited to this element .…”

Orbifold Kähler Groups related to Mapping Class groups

“…Proof. When i 1 = 2 this is the main result of ( [36], Thm.4.1) noting that their proof works for all p ≥ 5 not only for large enough p. For other values of i 1 = 0 this is contained in the proof of ( [25], Prop. 3.2, see also [24]).…”

“…The last item follows by observing that the proof given in [36] for i 1 = 2, can be adapted without any change to even p (see [25]).…”

“…The groups Mod(Σ g )/Mod(Σ g )[p] were introduced and studied in [23] and Mod(Σ 1 g )/ Mod(Σ 1 g )[p] in [25]. In [2] the authors proved that they are virtually Kähler groups by considering the Deligne-Mumford compactifications of the moduli spaces of curves for some level structures for which they are smooth projective varieties (see e.g.…”

On mapping class group quotients by powers of Dehn twists and their representations

“…Eventually we should note that the results obtained above for Γ g could in principle be extended to Γ r g,k . The case of Γ 1 g was first considered by Koberda-Santharoubane ( [42]) and further in [32,21]. In particular we have linear algebraic groups U 1 g,p , playing the role of U p and projective representations ρ p : Γ 1 g → P U 1 g,p .…”