Graph Immersions, Inverse Monoids, and Deck Transformations

Abstract

If $f : \tildeΓ \rightarrow Γ$ is a covering map between connected graphs, and $H$ is the subgroup of $π_1(Γ,v)$ 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 $π_1(Γ,v)$. 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 $L(Γ,v)$ 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 $f : \tildeΓ \rightarrow Γ$ may be extended to a cover $g : \tildeΔ \rightarrow Γ$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.