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Mathematics of Computation

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On split liftings with sectional complements


Authors: Aleksander Malnič and Rok Požar
Journal: Math. Comp.
MSC (2010): Primary 05C50, 05C85, 05E18, 20B40, 20B25, 20K35, 57M10, 68W05
DOI: https://doi.org/10.1090/mcom/3352
Published electronically: June 5, 2018
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Abstract: Let $ p\colon \tilde {X} \to X$ be a regular covering projection of connected graphs, where $ \hbox {{\rm CT}}_{\wp }$ denotes the group of covering transformations. Suppose that a group $ G \leq {\rm Aut} \,X$ lifts along $ \wp $ to a group $ \tilde {G} \leq {\rm Aut} \,\tilde {X}$. The corresponding short exact sequence $ {\rm id} \to \hbox {{\rm CT}}_{\wp } \to \tilde {G} \to G \to \rm {id}$ is split sectional over a $ G$-invariant subset of vertices $ \Omega \subseteq V(X)$ if there exists a sectional complement, that is, a complement $ \overline {G}$ to $ \hbox {{\rm CT}}_{\wp }$ with a $ \overline {G}$-invariant section $ \overline {\Omega } \subset V(\tilde {X})$ over $ \Omega $. Such lifts do not split just abstractly but also permutationally in the sense that they enable a nice combinatorial description.

Sectional complements are characterized from several viewpoints. The connection between the number of sectional complements and invariant sections on one side, and the structure of the split extension itself on the other, is analyzed. In the case when $ \hbox {{\rm CT}}_{\wp }$ is abelian and the covering projection is given implicitly in terms of a voltage assignment on the base graph $ X$, an efficient algorithm for testing whether $ \tilde {G}$ has a sectional complement is presented. Efficiency resides on avoiding explicit reconstruction of the covering graph and the lifted group. The method extends to the case when $ \hbox {{\rm CT}}_{\wp }$ is solvable.


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Additional Information

Aleksander Malnič
Affiliation: University of Ljubljana, PeF, Kardeljeva pl. 16, 1000 Ljubljana, Slovenia; and University of Primorska, IAM, Muzejski trg 2, 6000 Koper, Slovenia
Email: aleksander.malnic@guest.arnes.si

Rok Požar
Affiliation: University of Primorska, FAMNIT, Glagoljaška 8, 6000 Koper, Slovenia
Email: pozar.rok@gmail.com

DOI: https://doi.org/10.1090/mcom/3352
Keywords: Algorithm, Cayley voltages, covering projection, graph, group presentation, invariant section, lifting automorphisms, linear systems over the integers, split extension
Received by editor(s): May 13, 2017
Received by editor(s) in revised form: December 18, 2017
Published electronically: June 5, 2018
Additional Notes: The first author was supported in part by the Slovenian Research Agency, research program P1-0285 and research projects N1-0032, N1-0038, J1-5433, J1-6720, J1-7051.
The first author is the corresponding author
This work was supported in part by the Slovenian Research Agency, research program P1-0285 and research project J1-6720.
Article copyright: © Copyright 2018 American Mathematical Society

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