Negative Immersions and Finite Height Mappings

TL;DR

This paper characterizes when the mapping torus of a monomorphism between free groups has negative immersions, linking it to finite height and full irreducibility, and explores related properties and future directions.

Contribution

It establishes an equivalence between negative immersions, finite height, and full irreducibility for monomorphisms of free groups, providing new insights into their geometric and algebraic properties.

Findings

Negative immersions occur iff the monomorphism is fully irreducible.

Finite height of the subgroup corresponds to negative immersions.

The paper discusses related properties and potential future research directions.

Abstract

Given a monomorphism $\Psi:\mathcal{H}\rightarrow \mathcal{F}$ where $\mathcal{H}$ is a proper free factor of the free group $\mathcal{F}$, we show the associated mapping torus $X$ of $\Psi$ has negative immersions iff $\mathcal{H}$ has finite height in $\pi_1X$ iff $\Psi$ is fully irreducible. We survey related properties and discuss possible directions to pursue further.