Multiplied configurations induced by quasi difference sets (Q2657857)

In the paper under review, the authors construct different configurations induced by quasi difference sets. Let \(D\) be some fixed subset of a group \(G\). For our configuration, points are just elements of \(G\), and block (lines) are the images of \(D\) under left translations. If every non-zero element of \(G\) can be presented in exactly \(\lambda\) ways as a difference of two elements in \(D\), then \(D\) is called a difference set. In this way one obtains a \(\lambda\)-design. Difference sets with \(\lambda=1\) are called Singer (or planar) difference sets and induce a linear spaces. In order to get weaker geometries, one can admit sets with \(\lambda \in \{0,1\}\) and call them quasi difference sets. In the paper under review the authors consider configurations which can be defined with use of arbitrary quasi difference sets. Elementary properties of these structures are discussed, for instance the authors study the so-called satisfiability of Veblen, Pappus, and Desargues axioms. A special emphasis is imposed on structures which arise from groups decomposed into a cyclic group \(C_{k}\) and some other group \(G\). These structures can be seen as multiplied configurations -- series of cyclically inscribed configurations, each one isomorphic to the configuration associated with the group \(G\). On the other hand, this construction is just a special case of the operation of joining two structures, corresponding to the operation of the direct sum of groups. In some cases, the corresponding decomposition can be defined within the resulting join, in terms of the geometry of the considered structures. This definable decomposition enables to characterize the automorphism group of such a join. Some other techniques are used to determine the automorphism group of cyclically inscribed configuration. Roughly speaking, groups in question are semidirect products of some symmetric group and the group of translations of the underlying group. The technique of quasi difference sets can be used to produce new configuration. Many of them seem to be of a real geometrical interest on their own rights. In the last section, the authors apply their tools to get some new configurations arising from the well-known examples, for instance cyclically inscribed Pappus or Fano configurations, multiplied Pappus configurations, a power of cyclic projective planes, etc.