Paper 4, Section I, F

Geometry of Group Actions
Part II, 2010

Define loxodromic transformations and explain how to determine when a Möbius transformation

T:zaz+bcz+d with adbc=1T: z \mapsto \frac{a z+b}{c z+d} \quad \text { with } \quad a d-b c=1

is loxodromic.

Show that any Möbius transformation that maps a disc Δ\Delta onto itself cannot be loxodromic.