The geometry of wreath and semi-direct products

TL;DR

This paper investigates how geometric structures called coset geometries can be extended to include operations like twisting and wreath products, preserving key properties and enabling the construction of new regular polytopes and hypertopes.

Contribution

It extends the framework of coset geometries to include twisting and wreath products, demonstrating their preservation of important properties and applications to complex polytopes.

Findings

Twisting and wreath product operations extend to coset geometries.

These operations preserve flag-transitivity, residual-connectedness, and thinness.

Existence of regular polytopes and hypertopes for almost-simple groups with sporadic simple group socles.

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.