The existence of pyramidal Steiner triple systems over abelian groups

Abstract: A Steiner triple system STS$(v)$ is called $f$-pyramidal if it has an automorphism group fixing $f$ points and acting sharply transitively on the remaining ones. In this paper, we focus on the STSs that are $f$-pyramidal over some abelian group. Their existence has been settled only for the smallest admissible values of $f$, that is, $f=0,1,3$. In this paper, we complete this result and determine, for every $f>3$, the spectrum of values $(f,v)$ for which there is an $f$-pyramidal STS$(v)$ over an abelian group. This result is obtained by constructing difference families relative to a suitable partial spread.