Graph Immersions, Inverse Monoids, and Deck Transformations

If $f:\tilde{\mathrm{\Gamma }}\to \mathrm{\Gamma }$ is a covering map between connected graphs, and $H$ is the subgroup of ${\pi }_1(\mathrm{\Gamma },v)$ used to construct the cover, then it is well known that the group of deck transformations of the cover is isomorphic to $N(H)\mathrm{/}H$, where $N(H)$ is the normalizer of $H$ in ${\pi }_1(\mathrm{\Gamma },v)$. 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 $L(\mathrm{\Gamma },v)$ 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 $f:\tilde{\mathrm{\Gamma }}\to \mathrm{\Gamma }$ may be extended to a cover $g:\tilde{\mathrm{\Delta }}\to \mathrm{\Gamma }$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.