In the classic textbook Fusian Groups by Kadok, the first chapter, the author gives us several theorems that, in the upper half plane model of 2d hyperbolic space, PSL(2, R) is angle preserving, distance preserving, geodiscs preserving and area preserving. But these theorems are very tidy, precise, and carefully-given, the author didn't mention anything about the relationships in the other direction, eg
- is there other mapping than PSL(2, R) which is also angle preserving
- is there other mapping than PSL(2, R) which is also distance preserving
- is there other mapping than PSL(2, R) which is also geodiscs preserving
- is there other mapping than PSL(2, R) which is also area preserving
So in all, in hyperbolic and general settings, what the relationships of angle-preserving, distance-preserving, geodiscs-preserving and area-preserving are? Could somebody gives us a Venn diagram for these basic concepts?