Negative Immersions and Finite Height Mappings

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.