Generalize the Schwarz reflection principle to non-reflection wallpaper groups

Develop and prove a generalization of the Schwarz reflection principle that applies to wallpaper symmetry groups without reflection symmetries, such as the Euclidean wallpaper group with orbifold notation 2222 (four non-transitive order‑2 rotation centers), so that a conformal map from a hyperbolic fundamental polygon to the corresponding Euclidean fundamental polygon can be uniquely extended across the boundary to produce a conformal hyperbolization when classical reflection-based SRP is inapplicable.

Background

The paper’s algorithm constructs conformal hyperbolizations by mapping a Euclidean fundamental region to a hyperbolic one and extending the map globally. For reflection groups, the Schwarz reflection principle ensures that extension by symmetry coincides with analytic continuation, yielding a globally conformal map.

For groups lacking reflections (e.g., the orbifold 2222), the classical SRP cannot be used. The authors report promising numerical evidence suggesting that a suitable generalization of SRP could yield a unique conformal hyperbolization in this setting, but no formal proof is currently available.

References

In this case, the Neandertal algorithm requires further adaption, as the underlying symmetry group lacks reflections and the SRP cannot be applied. Nevertheless, it appears that a suitable generalization of the SRP can produce a unique conformal hyperbolization in this setting. Although no formal proof exists so far, numerical evidence looks promising.

— The Smooth Power of the "Neandertal Method" (2507.07569 - Montag et al., 10 Jul 2025) in Section 6, Subsection “2222 and bigcirc”