Constructing new geometries: a generalized approach to halving for hypertopes

Given a residually connected incidence geometry Γ that satisfies two conditions, denoted (B₁) and (B₂), we construct a new geometry H(Γ) with properties similar to those of Γ. This new geometry H(Γ) is inspired by a construction of Percsy, Percsy and Leemans [1]. We show how H(Γ) relates to the classical halving operation on polytopes, allowing us to generalize the halving operation to a broader class of geometries, that we call non-degenerate leaf hypertopes. Finally, we apply this generalization to cubic toroids in order to generate new examples of regular hypertopes.