Genus 3 mapping class groups are not Kähler

Abstract

We prove that finite index subgroups of genus 3 mapping class and Torelli groups that contain the group generated by Dehn twists on bounding simple closed curves are not Kähler. These results are deduced from explicit presentations of the unipotent (aka, Malcev) completion of genus 3 Torelli groups and of the relative completions of genus 3 mapping class groups. The main results follow from the fact that these presentations are not quadratic. To complete the picture, we compute presentations of completed Torelli and mapping class in genera ≥ 4; they are quadratic. We also show that groups commensurable with hyperelliptic mapping class groups and mapping class groups in genera ≤ 2 are not Kähler. ContentsTheorem 4. In genus 3, the Lie algebras t 3 and u 3 are isomorphic to the degree completions of the graded Lie algebras associated to their lower central series:Gr n LCS t 3 and u 3 ∼ = n≥1 Gr n LCS u 3

Homology of depth-graded motivic Lie algebras and koszulity

“…In §4.1, we prove one implication, and in §4.2, we show the converse implication. As mentioned in the Introduction, the results of this section are close to those of [17] …”

“…In Section 4, we show that a part of the homological conjecture for one of these Lie algebras (more precisely, the part predicting the values of the first and second homology groups) is equivalent to a presentation of the same Lie algebra. The proof of this result is close to the proof that any positively graded Lie algebra L has a presentation with generating space H 1 (L) and relation space H 2 (L), where H i (L) denotes the ith Lie algebra homology group of L with trivial coefficients ([17, Section 3]); the arguments from [17] are themselves analogues of those of [24,Chap. 2], in the pro-p group situation.…”

Formality properties of finitely generated groups and Lie algebras

“…Therefore, the diagram is commutative. Moreover, the right vertical arrow from (32) is an isomorphism by Quillen's isomorphism (28), while the lower horizontal arrow is an isomorphism by Theorem 8.2.…”

On the completion of the mapping class group of genus two

“…From this, it follows that Gr W • u g has possibly quadratic and cubic relations for 3 ≤ g < 6, and only quadratic relations for g ≥ 6. In [9], Hain determined all quadratic relations for g ≥ 4, and in [6] showed that for g ≥ 4, Gr W • u g is quadratically presented and determined the quadratic and cubic relations for g = 3.…”