Counting Fractional Linear Transformations

Question

This is a problem from Ahlfors' Complex analysis, Section 3.5.

In an obvious way, which we shall not try to make precise, a family of [Möbius] transformations depends on a certain number of real parameters. How many real parameters are there in the family of all [fractional] linear transformations? How many in the families of hyperbolic, elliptic, parabolic transformations? How many fractional linear transformations leave a given circle invariant?

I'm not really sure what he's getting at. I am wondering my thoughts below are in the general direction of what kind of reasoning he's looking for.

All fractional linear transformations

$Tz = \frac{az + b}{cz + d}$ looks like it has 4 complex numbers, so at first I thought there should be 8 real numbers involved. But, many collections give the same transformation, so if you normalize using the determinant, you're left with three degrees of freedom in complex numbers. So you need 6 real parameters.

Hyperbolic transformations

You can write these like $\frac{Tz - a}{Tz - b} = k \frac{z-a}{z-b}$, where $k$ is real, and $a$ and $b$ are complex. So you need 5 real numbers to specify one.

Elliptic transformations

Pretty much the same as hyperbolic, except $k = e^{i\theta}$. So also 5 parameters in this case.

Parabolic transformations

You can write these like $\frac{\omega}{Tz - a} = \frac{\omega}{z-a} + c$ where $\omega$ and $a$ are complex and $c$ is real. So again 5 real parameters.

Transformations taking a given circle to itself

Pich three points on the circle $z_1, z_2, z_3$. Transformations taking the circle to itself are in bijective correspondence with triples $w_1, w_2, w_3$ with $|w_i - c| = r$. So $w_i = c + r e^{ i \theta_i}$. One real parameter is required for each $w_i$, and so three real parameters are required to specify such a transformation.

Answer 1

Your answers are correct, except for the parabolic case. Only four real parameters are needed there: you could divide the equation by $c$, for example.

I can't speak for Ahlfors, but your reasoning is about what I would expect for this question.

Perhaps one can make it more geometric by focusing on fixed points a bit more. Elliptic and hyperbolic transformations have two fixed points, to specify which we need four real parameters. Then there's one more real parameter to specify by how much we rotate other points (elliptic) or slide them from one fixed point to another (hyperbolic).

For a parabolic transformation, there is only one fixed point (2 parameters), and if it's moved to infinity, we get to pick the complex number by which to translate (2 more parameters).

Incidentally, one sees from the above that "most" FLT are loxodromic: the union of the aforementioned 4- and 5- parameter families has measure zero in the 6-parameter family of all fractional linear transformations.