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Graph Immersions, Inverse Monoids, and Deck Transformations
If f:Γ~→Γ is a covering map between connected graphs, and H is the subgroup of π1(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 π1(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:Γ~→Γ may be extended to a cover g:Δ~→Γ in such a way that all deck transformations of f are restrictions of deck transformations of g.
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