On the stabilisation height of fibre surfaces in $S^3$

TL;DR

This paper investigates the minimal number of Hopf plumbing operations required to stabilize fibre surfaces in the 3-sphere, revealing that certain families of fibre surfaces have unbounded stabilisation height.

Contribution

It demonstrates that families of fibre surfaces related by iterated Stallings twists can have arbitrarily large stabilisation heights, highlighting new complexity in fibre surface stabilization.

Findings

Families related by Stallings twists have unbounded stabilisation height

Stabilisation height can grow arbitrarily large

Provides insights into the complexity of fibre surface stabilization

Abstract

The stabilisation height of a fibre surface in the 3-sphere is the minimal number of Hopf plumbing operations needed to attain a stable fibre surface from the initial surface. We show that families of fibre surfaces related by iterated Stallings twists have unbounded stabilisation height.