Negative Immersions and Finite Height Mappings

Given a monomorphism $\mathrm{\Psi }:\mathcal{H}\to \mathcal{F}$ where $\mathcal{H}$ is a proper free factor of the free group $\mathcal{F}$, we show the associated mapping torus $X$ of $\mathrm{\Psi }$ has negative immersions iff $\mathcal{H}$ has finite height in ${\pi }_{1}X$ iff $\mathrm{\Psi }$ is fully irreducible. We survey related properties and discuss possible directions to pursue further.