Extending powers of pseudo-Anosovs

TL;DR

This paper demonstrates that the power needed for a pseudo-Anosov map to partially extend into a 3-manifold's interior is unbounded by constructing specific examples where the minimal power varies.

Contribution

It proves that the minimal power for partial extension of pseudo-Anosov maps is not bounded, providing explicit constructions of such maps with variable extension powers.

Findings

The required power for partial extension can grow arbitrarily large.

Explicit examples of pseudo-Anosov maps with variable extension powers.

Partial extension depends on the specific map and manifold, not just on the map itself.

Abstract

Biringer, Johnson, and Minsky showed that a pseudo-Anosov map on a boundary component of an irreducible 3-manifold has a power that partially extends to the interior if and only if the (un)stable laminations of $f$ is an $\mathbb{R}$-projective limit of meridians. We prove that the power required for a pseudo-Anosov map to partially extend is not universally bounded. We construct a family of pseudo-Anosov maps $f_i$ for all $i=1,2...$ on a boundary component of a family of irreducible 3-manifolds $M_i$ such that $f_i^i$ partially extends to the interior of $M_i$ but $f_i^j$ does not for $j<i$.