On the split structure of lifted groups

Aleksander Malnič ; Rok Požar

Ars Mathematica Contemporanea, Tome 10 (2016) no. 1, pp. 113-134.

Résumé

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.

Détail

DOI : 10.26493/1855-3974.670.306

Keywords: Algorithm, abelian cover, Cayley voltages, covering projection, graph, group extension, group presentation, lifting automorphisms, linear systems over the integers, semidirect product.