Finally, we will discuss some further developments of the theories described in this book. For recent references and more details the reader is referred to Bers [1972], Thurston [1982], Beardon [1983].

In Chapter 2 we investigated the group PSL(2, ℝ) in detail and in Chapter 5 we considered discrete subgroups of PSL(2, ℝ) with particular reference to plane hyperbolic geometry. As we have already mentioned, the connection between hyperbolic geometry and PSL(2, ℝ) was formulated by Poincaré and published in 1882. In a paper published a year later Poincaré studied discrete subgroups of PSL(2,ℂ) using 3-dimensional hyperbolic geometry (Poincaré [1883]). As the topology of 3-dimensional manifolds was so little understood at the end of the nineteenth century this interesting connection between 3-dimensional manifolds and discrete subgroups of PSL(2,ℂ) was rather neglected. In recent years 3-dimensional topology has advanced considerably and this connection is proving to be of great importance.

We consider ℝ3∪{∞} as the one-point compactification of ℝ3, (see §1.2). We identify x + iy∈ℂ with the point (x,y,0)∈ℝ3 so that ℂ∪{∞} is a subset of ℝ3∪{∞}; we also denote upper half 3-space {(x,y,u)∈ℝ3|u>0 by U3. We now describe how elements of PSL(2,ℂ) act as directly conformal (angle- and orientation-preserving) transformations of U3.

If T∈PSL(2,ℂ) then by exercises 2E - G, T is a product of an even number of inversions in circles (where this includes reflections in lines, §2.7).