By T. K. Carne

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**Extra info for Geometry and Groups [Lecture notes]**

**Example text**

Hence M is the quotient PSL(2, Z) of the matrix group SL(2, Z) by {I, −I}. The modular group acts as a group of hyperbolic isometries on the upper half-plane R2+ . This group is certainly discrete because the integers are discrete. We wish to find a fundamental set for the modular group and the corresponding tessellation. Consider the orbit Ω of a point w ∈ R2+ . For a transformation T : z → az+b cz+d in the modular group we have 1 (aw + b)(cw + d) = Im(w) . Im(T (w)) = Im |cw + d|2 |cw + d|2 Hence, ⇔ Im(T (w)) > Im(w) 1 > |cw + d| .

It is now fairly simple to visualise how M¨obius transformations act on the Riemann sphere. An elliptic transformation is conjugate to z → eiθ z. This rotates the sphere fixing 0 and ∞. Each point is moved along a circle. A hyperbolic transformation is conjugate to z → kz for k > 1. This moves points along arcs of circles from one fixed point towards the other. A loxodromic transformation is conjugate to z → kz for k ∈ / R. This moves points along logarithmic spirals away from one fixed point and towards the other.

A loxodromic transformation is conjugate to z → kz for k ∈ / R. This moves points along logarithmic spirals away from one fixed point and towards the other. Finally, a parabolic transformation is conjugate to z → z + 1. This maps points along a circle through the single fixed point. The points are mapped away on one side and towards on the other. The pictures below illustrate this. 2 Inversion Let Γ be a circle on the Riemann sphere. Two distinct points z, z are inverse points for Γ if every circle orthogonal to Γ through z also passes through z .

### Geometry and Groups [Lecture notes] by T. K. Carne

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