On the split structure of lifted groups - Malnič, Aleksander (Ars mathematica contemporanea, 2016)
Let ?$\wp \colon \tilde{X} → X$? be a regular covering projection of connected graphs with the group of covering transformations ?$\rm{CT}_\wp$? being abelian. Assuming that a group of automorphisms ?$G \le \rm{Aut} X$? lifts along $\wp$ to a group ?$\tilde{G} \le \rm{Aut} \tilde{X}$?, the problem whether the corresponding exact sequence ?$\rm{id} → \rm{CT}_\wp → \tilde{G} → G → \rm{id}$? 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 ?$\tilde{G}$? nor the action ?$G→ \rm{Aut} \rm{CT}_\wp$? nor a 2-cocycle ?$G × G → \rm{CT}_\wp$?, are given. Explicitly constructing the cover ?$\tilde{X}$? together with ?$\rm{CT}_\wp$? and ?$\tilde{G}$? as permutation groups on ?$\tilde{X}$? is time and space consuming whenever ?$\rm{CT}_\wp$? 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 ?$\rm{CT}_\wp$? is elementary abelian.
Keywords: abelian cover; algorithm; Cayley voltages; covering projection; graph; group extension; group presentation; lifting automorphisms; linear systems over the integers; semidirect product