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GRAPH IMMERSIONS, INVERSE MONOIDS AND DECK TRANSFORMATIONS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 111 / Issue 1 / August 2021
- Published online by Cambridge University Press:
- 02 March 2020, pp. 37-55
- Print publication:
- August 2021
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- Article
- Export citation
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If
$f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ is a covering map between connected graphs, and 
$H$ is the subgroup of 
$\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$ 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 
$\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6E4},v)$. 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 
$L(\unicode[STIX]{x1D6E4},v)$ 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 
$f:\tilde{\unicode[STIX]{x1D6E4}}\rightarrow \unicode[STIX]{x1D6E4}$ may be extended to a cover 
$g:\tilde{\unicode[STIX]{x1D6E5}}\rightarrow \unicode[STIX]{x1D6E4}$ in such a way that all deck transformations of 
$f$ are restrictions of deck transformations of 
$g$.
