Lifting constructions for semi-direct products

Develop a lifting construction for semi-direct products that, analogously to known lifting methods for direct products, converts an aperiodic monotile in a semi-direct product group (such as the Coxeter symmetry group Γ of the Semikitegrid) into an explicit weakly aperiodic monotile in Euclidean settings such as R^d or Z^d, thereby enabling explicit realizations comparable to results that are currently available only for direct products.

Background

The paper constructs an explicit aperiodic group monotile (the Roach) in the Coxeter group Γ, which is a semidirect extension closely related to Z^2. Known lifting operations can transfer aperiodic monotiles from direct product groups to Euclidean spaces, enabling explicit constructions in R^d or Z^d.

Because Γ arises as a semidirect product rather than a direct product, the standard lifting approach does not apply. Extending the lifting framework to semidirect products would bridge the gap between the group-theoretic construction and explicit Euclidean monotiles, potentially yielding explicit versions comparable to existing high-dimensional results.