On split liftings with sectional complements

Aleksander Malnič (Author), Rok Požar (Author)

Abstract

Let ▫$ p\colon \tilde {X} \rightarrow X$▫ be a regular covering projection of connected graphs, where ▫$ {\mathrm{CT}}_{\mathcal P}$▫ denotes the group of covering transformations. Suppose that a group ▫$ G \leq \mathrm{Aut} \,X$▫ lifts along ▫$\mathcal P$▫ to a group ▫$ \tilde {G} \leq \mathrm{Aut} \,\tilde {X}$▫. The corresponding short exact sequence ▫$ \mathrm{id} \rightarrow \mathrm {CT}_{\mathcal P} \rightarrow \tilde {G} \rightarrow G \rightarrow \mathrm{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}$▫ to ▫$ \mathrm{CT}_{\mathcal P}$▫ with a ▫$ \overline {G}$▫-invariant section ▫$ \overline {\Omega } \subset V(\tilde {X})$▫ over ▫$ \Omega $▫. Such lifts do not split just abstractly but also permutationally in the sense that they enable a nice combinatorial description. Sectional complements are characterized from several viewpoints. The connection between the number of sectional complements and invariant sections on one side, and the structure of the split extension itself on the other, is analyzed. In the case when ▫$ \mathrm{CT}_{\mathcal P}$▫ is abelian and the covering projection is given implicitly in terms of a voltage assignment on the base graph ▫$ X$▫, an efficient algorithm for testing whether ▫$ \tilde {G}$▫ has a sectional complement is presented. Efficiency resides on avoiding explicit reconstruction of the covering graph and the lifted group. The method extends to the case when ▫$\mathrm{CT}_{\mathcal P}$▫ is solvable.

Keywords

algorithm;Cayley voltages;covering projection;graph;group presentation;invariant section;lifting automorphisms;linear systems over the integers;split extension;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	519.17
COBISS:	1540135364
ISSN:	0025-5718
Views:	2277
Downloads:	175
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Pages:	str. 983-1005
Volume:	ǂVol. ǂ88
Issue:	ǂno. ǂ316
Chronology:	March 2019
DOI:	10.1090/mcom/3352
ID:	10908755