Equivalence of extension by symmetry and analytic continuation beyond reflection groups

Prove that for wallpaper symmetry groups other than pure reflection groups (including the Euclidean wallpaper group with orbifold notation 2222), extension of a conformal map beyond the boundary of the fundamental region by applying the group’s isometries coincides with analytic continuation, analogously to the Schwarz reflection principle for reflection groups, thereby ensuring that hyperbolizations produced via extension by symmetry are conformal.

Background

The authors distinguish two ways to extend conformal maps beyond the fundamental region: by symmetry (using group operations) and by analytic continuation. For reflection groups these coincide via the Schwarz reflection principle.

They conjecture that this equivalence also holds for other symmetry groups lacking reflections. Establishing this result would guarantee that their symmetry-based extension algorithm yields conformal hyperbolizations for these groups.