A generating set for the Johnson kernel

TL;DR

This paper proves that the Johnson kernel of a hyperbolic surface is generated by Dehn twists about separating simple closed curves bounding specific types of subsurfaces, clarifying its generating set.

Contribution

It establishes a generating set for the Johnson kernel using Dehn twists about particular separating curves, advancing understanding of its algebraic structure.

Findings

Johnson kernel generated by twists about genus 1 or 2 subsurfaces

Includes twists about genus 1 minus one point and disc minus two points

Provides explicit generating set for the Johnson kernel

Abstract

For a connected orientable hyperbolic surface $S$ without boundary and of finite topological type, the Johnson kernel ${\mathcal K}(S)$ is the subgroup of the mapping class group of $S$ generated by Dehn twists about separating simple closed curves on $S$. We prove that ${\mathcal K}(S)$ is generated by the Dehn twists about separating simple closed curves on $S$ bounding either: a closed subsurface of genus $1$ or $2$; a closed subsurface of genus $1$ minus one point; a closed disc minus two points.