Geometry | Part IB, 2001

Describe the hyperbolic lines in the upper half-plane model HH of the hyperbolic plane. The group G=SL(2,R)/{±I}G=\mathrm{SL}(2, \mathbb{R}) /\{\pm I\} acts on HH via Möbius transformations, which you may assume are isometries of HH. Show that GG acts transitively on the hyperbolic lines. Find explicit formulae for the reflection in the hyperbolic line LL in the cases (i) LL is a vertical line x=ax=a, and (ii) LL is the unit semi-circle with centre the origin. Verify that the composite TT of a reflection of type (ii) followed afterwards by one of type (i) is given by T(z)=z1+2aT(z)=-z^{-1}+2 a.

Suppose now that L1L_{1} and L2L_{2} are distinct hyperbolic lines in the hyperbolic plane, with R1,R2R_{1}, R_{2} denoting the corresponding reflections. By considering different models of the hyperbolic plane, or otherwise, show that

(a) R1R2R_{1} R_{2} has infinite order if L1L_{1} and L2L_{2} are parallel or ultraparallel, and

(b) R1R2R_{1} R_{2} has finite order if and only if L1L_{1} and L2L_{2} meet at an angle which is a rational multiple of π\pi.

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