174 31. HYPERBOLIC GEOMETRY AND 3-MANIFOLDS¢(Mi::ib(n-1)) is exactly the subgroup of Mi::ibius transformations of !Rn leaving un
invariant.
By inspecting the metric gu, we see that isometries of un include f(x) and g(x)
defined by
(1) f(x) = kx, where k is a positive real number,
(2)
(31.2) g(w,xn) = (Aw+a,xn),
where w = (x 1 , ... ,Xn-1) E !Rn-l, Xn E IR>o, A E O(n-1), and a E !Rn-^1.It can be shown that the set of all isometries of un is the set of compositions
of the functions f, g , and lo, 1 described previously. So the group Mi::ib(n - 1) may
be identified with the isometry group of hyperbolic space.
Let Isom(IHin) and Isom+ (IHin) be the groups of isometries and orientation-
preserving isometries of hyperbolic n -space IHin. We can summarize the discussion
as
LEMMA 31.5. Using the un model of hyperbolic space, we see that
Isom (IHin) S'! Mi::ib(n - 1)= { x H .\A (i (x)) + b: .\ E IR>o, A E 0 (n - 1), i =id or ly,r, b E !Rn-l}.
Moreover, Mi::ib+(n - 1) S'! Isom+ (JHin) consists of the elements of Mi::ib(n - 1)
where i = id and A E SO(n - 1) or i is an inversion about a sphere and A E
O(n - 1) - SO(n - 1).We now give a more detailed description of the isometry group of 3-dimensional
hyperbolic space IHI^3. Two-dimensional Mi::ibius transformations can be expressed
nicely in terms of the linear fractional transformations and reflection z H z; i.e.,
each orientation-preserving Mi::ibius transformation of the complex plane is of the
form
az +b
cPM: Z f----t --dcz+ 'where M = ( ~ ~ ) ESL (2, C); i.e., a , b, c, d EC satisfy ad - be= l. Note that:
(i) cPM' = ¢M if and only if M' = ±M.
(ii) The map from PSL (2, q ~SL (2 , q /{±id} to Mi::ib+(2) defined by
[M] f----t cPM
is a Lie group isomorphism.
As a consequence, we have the isomorphisms of groups
(31.3)where matrix multiplication in SL (2, q induces multiplication in its Z 2 -quotient
PSL(2,C).
Similarly, our earlier discussion has shown that we have the isomorphisms
(31.4) Isom+ (IHI^2 ) S'! Mi::ib+(1) S'! PSL (2, IR) ~SL (2 , IR)/{± id},
where PSL(2, IR) acts on the upper half-plane by linear fractional transformations