Three-Point Lemma for Möbius Transformations
If a Möbius transformation fixes three distinct points of , then it is the identity.
Lemma 5.6 (Three-Point Lemma For )
If fixes three distinct points of , then .
Proof
If one fixed point is , then and the other fixed points satisfy the linear equation . Two distinct finite roots force and . If no fixed point is , then the fixed points satisfy the quadratic equation . Three distinct roots force , , and . In either case, the transformation is the identity.
Related
Stated in
- Lemma 5.6 (Three-Point Lemma For )§5.2 The Möbius Group