Graph Immersions, Inverse Monoids, and Deck Transformations

If $f : \tilde{\Gamma} \rightarrow \Gamma$ is a covering map between connected graphs, and $H$ is the subgroup of $\pi_1(\Gamma,v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)/H$, where $N(H)$ is the normalizer of $H$ in $\pi_1(\Gamma,v)$. We show that an entirely analogous result holds for immersions between connected graphs, where the subgroup $H$ is replaced by the closed inverse submonoid of the inverse monoid $L(\Gamma,v)$ used to construct the immersion. We observe a relationship between group actions on graphs and deck transformations of graph immersions. We also show that a graph immersion $f : \tilde{\Gamma} \rightarrow \Gamma$ may be extended to a cover $g : \tilde{\Delta} \rightarrow \Gamma$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.