Graph Immersions, Inverse Monoids, and Deck Transformations

DOI10.1017/S1446788720000087arXiv1903.07203MaRDI QIDQ6315763

Corbin Groothuis, J. C. Meakin

Publication date: 17 March 2019

Abstract: If $f:ildeGammaightarrowGamma$ is a covering map between connected graphs, and $H$ is the subgroup of $p{i}_1(Gamma,v)$ 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 $p{i}_1(Gamma,v)$. 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 $f:ildeGammaightarrowGamma$ may be extended to a cover $g:ildeDeltaightarrowGamma$ in such a way that all deck transformations of $f$ are restrictions of deck transformations of $g$.

Mathematics Subject Classification ID

Covering spaces and low-dimensional topology (57M10) Free semigroups, generators and relations, word problems (20M05) Inverse semigroups (20M18)

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