On the split structure of lifted groups
DOI:
https://doi.org/10.26493/1855-3974.670.306Keywords:
Algorithm, abelian cover, Cayley voltages, covering projection, graph, group extension, group presentation, lifting automorphisms, linear systems over the integers, semidirect product.Abstract
Let wp: tildeX -> X be a regular covering projection of connected graphs with the group of covering transformations CT_wp being abelian. Assuming that a group of automorphisms G <= Aut (X) lifts along wp to a group tildeG <= Aut(tildeX), the problem whether the corresponding exact sequence id -> CT_wp -> tildeG -> G -> id splits is analyzed in detail in terms of a Cayley voltage assignment that reconstructs the projection up to equivalence.
In the above combinatorial setting the extension is given only implicitly:
neither tildeG nor the action G -> Aut CT_wp nor a 2-cocycle GxG -> CT_wp, are given. Explicitly constructing the cover tildeX together with CT_wp and tildeG as permutation groups on tildeX is time and space consuming whenever CT_wp is large; thus, using the implemented algorithms (for instance, HasComplement in Magma) is far from optimal. Instead, we show that the minimal required information about the action and the 2-cocycle can be effectively decoded directly from voltages (without explicitly constructing the cover and the lifted group); one could then use the standard method by reducing the problem to solving a linear system of equations over the integers. However, along these lines we here take a slightly different approach which even does not require any knowledge of cohomology. Time and space complexity are formally analyzed whenever CT_wp is elementary abelian.
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