The geometry of wreath and semi-direct products

Abstract

Coset geometries are incidence geometries constructed from a group $G$ and a system of subgroups $(G_i)_{i \in I}$ of subgroups of $G$. For any algebraic group operation, it is then natural to wonder whether it can be extended to the framework of coset geometries. This has been achieved in the case of the halving (\cite{halving}) and in the case of free (amalgamated) products, HNN-extensions, and semi-direct products (\cite{piedade2025group}). In this article, we explore more deeply two operations related to semi-direct products: the twisting and the wreath product. We show that these operations extend to coset geometries in such a way that they preserve key properties, such as flag-transitivity, residual-connectedness and being thin. In particular, we can apply twistings and wreath products to polytopes and hypertopes. Doing so, we show that there exists regular polytopes and hypertopes for almost-simple group with socle a sporadic simple group.