Elementary Abelian Covers of Graphs

Abstract

Let \cal CG(X) be the set of all (equivalence classes of) regular covering projections of a given connected graph X along which a given group G ≤ Aut X of automorphisms lifts. There is a natural lattice structure on \cal CG(X), where ℘1 ≤ ℘2 whenever ℘2 factors through ℘1. The sublattice \cal CG(℘) of coverings which are below a given covering ℘ : X˜ → X naturally corresponds to a lattice \cal NG(℘) of certain subgroups of the group of covering transformations. In order to study this correspondence, some general theorems regarding morphisms and decomposition of regular covering projections are proved. All theorems are stated and proved combinatorially in terms of voltage assignments, in order to facilitate computation in concrete applications. For a given prime p, let \cal CGp(X) ≤ \cal CG(X) denote the sublattice of all regular covering projections with an elementary abelian p-group of covering transformations. There is an algorithm which explicitly constructs \cal CGp(X) in the sense that, for each member of \cal CGp(X), a concrete voltage assignment on X which determines this covering up to equivalence, is generated. The algorithm uses the well known algebraic tools for finding invariant subspaces of a given linear representation of a group. To illustrate the method two nontrival examples are included.