Useful properties of Mobius functions

This node is intended to serve as a list of various properties of möbius functions that seem useful, and as an index to those nodes that mention them. I hope someday to write up all these links...

Please node more mathematics by adding your own favourite properties in writeups below!

General properties

• matrix representation of möbius functions
• connection between möbius functions and PSL2

Möbius functions with complex coefficients

• automorphisms of the unit disk (these are also the isometries of the hyperbolic plane, when represented by Poincaré's disk model).
• automorphisms of the half plane (these are also the isometries of the hyperbolic plane, when represented by Poincaré's half plane model).
• In the Schwarz lemma, if we drop the requirement that the function fix 0, and demand merely a function from the disk to itself, by applying möbius functions we may reach the original formulation; the extremal case becomes a general möbius function rather than just a rotation.

Möbius functions with real coefficients

• monotonicity of real möbius functions
• asymptotes of real möbius functions

Möbius functions with integer coefficients

• There is a deep connection between möbius functions and continued fractions; all the algorithms used to build continued fractions relate to sets of generators of PSL2(Z) via möbius functions.
• free subgroup of PSL2(Z)