07/23/2022, 07:46 PM
I tested also combinations like index 1 and 2. They have a fundamental region however the image is not a region (\(b^z\) not injective).
These are the fundamental regions, they look odd already.
but with the mapping of the fundamental region you can not see much anymore, only that it is not injective:
So the conjecture here would be that only the fixed point combinations (0,1) and (0,-1) have fundamental regions with \(b^z\) injective on it.
More precisely (0,1) only in the case \(b\) is in the upper half plane and (0,-1) only in the case of the lower half plane.
These are the fundamental regions, they look odd already.
but with the mapping of the fundamental region you can not see much anymore, only that it is not injective:
So the conjecture here would be that only the fixed point combinations (0,1) and (0,-1) have fundamental regions with \(b^z\) injective on it.
More precisely (0,1) only in the case \(b\) is in the upper half plane and (0,-1) only in the case of the lower half plane.
