Geometry | Part IB, 2002

Let H={x+iyC:y>0}\mathbb{H}=\{x+i y \in \mathbb{C}: y>0\}, and let H\mathbb{H} have the hyperbolic metric ρ\rho derived from the line element dz/y|d z| / y. Let Γ\Gamma be the group of Möbius maps of the form z(az+b)/(cz+d)z \mapsto(a z+b) /(c z+d), where a,b,ca, b, c and dd are real and adbc=1a d-b c=1. Show that every gg in Γ\Gamma is an isometry of the metric space (H,ρ)(\mathbb{H}, \rho). For zz and ww in H\mathbb{H}, let

h(z,w)=zw2Im(z)Im(w)h(z, w)=\frac{|z-w|^{2}}{\operatorname{Im}(z) \operatorname{Im}(w)}

Show that for every gg in Γ,h(g(z),g(w))=h(z,w)\Gamma, h(g(z), g(w))=h(z, w). By considering z=iyz=i y, where y>1y>1, and w=iw=i, or otherwise, show that for all zz and ww in H\mathbb{H},

coshρ(z,w)=1+zw22Im(z)Im(w)\cosh \rho(z, w)=1+\frac{|z-w|^{2}}{2 \operatorname{Im}(z) \operatorname{Im}(w)}

By considering points i,iyi, i y, where y>1y>1 and s+its+i t, where s2+t2=1s^{2}+t^{2}=1, or otherwise, derive Pythagoras' Theorem for hyperbolic geometry in the form coshacoshb=coshc\cosh a \cosh b=\cosh c, where a,ba, b and cc are the lengths of sides of a right-angled triangle whose hypotenuse has length cc.

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