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{\bf Question}

Let

$$T(z)=\frac{az+b}{cz+d} \hspace{0.3in} a,b,c,d,\in {\bf R},
\hspace{0.3in} ad-bc=1$$

be a real Mobius transformation.  Show that $T$ maps the upper
half plane to itself.  Prove that the hyperbolic arc-length
defined by

$$ds=\frac{|dz|}{y} \hspace{0.5in} (z=x+iy)$$

is invariant under all real Mobius transformations.

Show that $6+4i$ and $7+3i$ are at the same Euclidean distance
from the point 3, and hence determine the hyperbolic line which
passes through $6+4i$ and $7+3i$.  By finding a Mobius
transformation which maps this hyperbolic line to the imaginary
axis compute the hyperbolic distance from $6+4i$ to $7+3i$.

\vspace{0.25in}

{\bf Answer}

1st bit only in this years course

$T$ maps the real axis to the real axis

im$\ds T(i)={\rm im}\frac{ai+b}{ci+d}={\rm
im}\frac{(ai+b)(-ci+d)}{c^2+d^2}=\frac{ad-bc}{c^2+d^2}>0$

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