| course |
course_year |
question_number |
tags |
title |
year |
Geometry |
IB |
27 |
|
Paper 2, Section II, G |
2018 |
For any matrix
$$A=\left(\begin{array}{ll}
a & b \\
c & d
\end{array}\right) \in S L(2, \mathbb{R})$$
the corresponding Möbius transformation is
$$z \mapsto A z=\frac{a z+b}{c z+d},$$
which acts on the upper half-plane $\mathbb{H}$, equipped with the hyperbolic metric $\rho$.
(a) Assuming that $|\operatorname{tr} A|>2$, prove that $A$ is conjugate in $S L(2, \mathbb{R})$ to a diagonal matrix $B$. Determine the relationship between $|\operatorname{tr} A|$ and $\rho(i, B i)$.
(b) For a diagonal matrix $B$ with $|\operatorname{tr} B|>2$, prove that
$$\rho(x, B x)>\rho(i, B i)$$
for all $x \in \mathbb{H}$ not on the imaginary axis.
(c) Assume now that $|\operatorname{tr} A|<2$. Prove that $A$ fixes a point in $\mathbb{H}$.
(d) Give an example of a matrix $A$ in $S L(2, \mathbb{R})$ that does not preserve any point or hyperbolic line in $\mathbb{H}$. Justify your answer.