Quasi-homomorphisms on mapping class groups vanishing on a handlebody group

TL;DR

This paper constructs numerous quasi-homomorphisms on mapping class groups that vanish on handlebody subgroups, leading to counterexamples to a conjecture and revealing infinite quasi-invariants in three-manifold splittings.

Contribution

It introduces infinitely many quasi-homomorphisms on mapping class groups that vanish on handlebody subgroups, disproving Reznikov's conjecture.

Findings

Disproved Reznikov's conjecture on bounded width in Heegaard splittings.

Established existence of infinitely many linearly independent quasi-invariants.

Constructed explicit quasi-homomorphisms with specific vanishing properties.

Abstract

We construct infinitely many linearly independent quasi-homomorphisms on the mapping class group of a Riemann surface with genus at least two which vanish on a handlebody subgroup. As a corollary, we disprove a conjecture of Reznikov on bounded width in Heegaard splittings. Another corollary is that there are infinitely many linearly independent quasi-invariants on the Heegaard splittings of three-manifolds.