Sectional split extensions arising from lifts of groups - Požar, Rok (Ars mathematica contemporanea, 2013)
Covering techniques have recently emerged as an effective tool used for classification of several infinite families of connected symmetric graphs. One commonly encountered technique is based on the concept of lifting groups of automorphisms along regular covering projections ?$\wp \colon \tilde{X} → X$?. Efficient computational methods are known for regular covers with cyclic or elementary abelian group of covering transformations CT?$(\wp)$?. In this paper we consider the lifting problem with an additional condition on how a group should lift: given a connected graph ?$X$? and a group ?$G$? of its automorphisms, find all connected regular covering projections ?$\wp \colon \tilde{X} → X$? along which ?$G$? lifts as a sectional split extension. By this we mean that there exists a complement ?$\overline{G}$? of CT?$(\wp)$? within the lifted group ?$\tilde{G}$? such that ?$\overline{G}$? has an orbit intersecting each fibre in at most one vertex. As an application, all connected elementary abelian regular coverings of the complete graph ?$K_4$? along which a cyclic group of order 4 lifts as a sectional split extension are constructed.
Keywords: covering projection; graph; group extension; lifting automorphisms; teorija grafov; voltage assignment