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GRAPH IMMERSIONS, INVERSE MONOIDS AND DECK TRANSFORMATIONS

Part of: Semigroups Low-dimensional topology

Published online by Cambridge University Press:  02 March 2020

CORBIN GROOTHUIS  and
JOHN MEAKIN
CORBIN GROOTHUIS
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, NE68588, USA e-mail: [email protected]
JOHN MEAKIN*
Affiliation:
Department of Mathematics, University of Nebraska, Lincoln, NE68588, USA
*
e-mail: [email protected]
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Abstract

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 $g$.

Keywords

graph immersiondeck transformationfree inverse monoid

MSC classification

Primary: 20M18: Inverse semigroups
Secondary: 57M10: Covering spaces
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 1 , August 2021 , pp. 37 - 55
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by B. Martin

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