As another example, if 0: G —*H is a homomorphism of the group G onto the group H, then the kernel K of B is a normal subgroup of G and the quotient group G/K is isomorphic to H. For each y in G, the mapping x i—*xy homeomorphism of G onto itself: the same holds for the mapping x i—* yx. Much of the material in this chapter can be written in terms of n x n complex matrices.
The center of a group is a set of elements that communicate with each element of the group. In this section we will restrict ourselves to subgroups of the topological group GL(2, C). This completes the proof of Theorem 3.1.3, since every isometry constructed with a corresponding reflection is of the form E.
Since an orthogonal matrix conserves lengths, it is clear that any given shape is an isometry. A useful substitute for conformality is the elegant concept of the inverse product E') of two spheres and E'. Now we will focus on the operation of the Poincaré expansion in First, if the reflection in the sphere is S(ä, r), but e then according to (3.1.5),.
It is a direct consequence of this invariance that the Poincare extension of every in is an isometry of the space endowed with the Riemannian metric p given by.
Let 4 be a Möbius transformation with = 0 and
PROOF. We only sketch the proof, as the interested reader can find a proof of the Arzel-Ascoli theorem elsewhere in the literature. By deleting the last number (which obviously does not affect the result) we can assume that for each n, i and j(i j). By these definitions, g is a 1-1 mapping of C onto itself. Moreover, g1 is of the same form.
These facts (which are left to the reader to verify) show that the class of maps of the form (4.1.2) is a group under the usual composition of functions. We can now review Theorems 2.5.1 and 2.5.2 in light of the geometric action of Möbius transformations. Of course, Theorem 4.2.2 shows that the classical symmetry groups of the regular solids (embedded in B3) correspond to the finite subgroups of SU(2, C): indeed, each rotation of B3 is represented by a Möbius g derived from a matrix in SU(2, C) and the symmetry groups can be realized as finite Mãbius groups.
EVIDENCE. The statements in (i) remain invariant under conjugation, so we can assume that in terms of matrices in SL(2, C). The map g a is a homomorphism of
E We end this section with a discussion of the iterates of a Möbius
In this case G is conjugate to a subgroup of .A where each element fixes and then has the form z az + b. In this case G is conjugate to a subgroup of .A1 where each element leaves {0, invariant and therefore is of the form. By conjugation, we can assume that each element of G fixes oo and so is of the form z H+ + /3.
First we suppose that each element of G captures both 0 and and is therefore of the form. It is now very easy to see that the multiplicative group 8(G) has the form. Then the sum of the areas of g(N), measured in the chord metric, converges to a maximum of 471 (the chord area of C) and it is only necessary to estimate this area of g(N).
The proof of Theorem 5.4.5 requires details of the geometry of the action of loxodromic and elliptic elements. It follows that 9 is defined independently of the choice of and is then regarded as the angle between y and at x. In all other cases, R has the form A/G, where G acts discontinuously in A and has no elliptical elements.
A Riemann surface R is said to be of hyperbolic type if it is of the form A/G where G acts in A. In this way we can talk about the hyperbolic metric on R and thus calculate lengths and areas on R. The union of the hyperbolic plane and the circle at infinity is called the closed hyperbolic.
We close this section with some brief remarks about the metric topology of the hyperbolic plane. A subgroup E of the hyperbolic plane is said to be convex if and only if for each . This maps the Poincare model geodesic (A, p) to Euclidean segments in A and so a subset E of A is convex in the Poincare model if and only if F(E) is convex in the Euclidean sense.
This region is convex because it complements the union of all half-planes of the shape. We will allow some or all of the vertices of a triangle to be on the infinite circle.
D= 1 +2ABC—(A2+B2+C2)
This is the last section on hyperbolic trigonometry, and we leave most of the details to the reader. This gives a subdivision of the entire hyperbolic plane into a finite number of non-overlapping convex polygons (convex since each is the intersection of half-planes). We now consider only the polygons P1 of the subdivision that lie in the original polygon P.
Saccheri in his study of the parallel postulate and is known as the Saccheri quadrilateral. The angle 0, 0 < 6 < it, between crossed geodesics can be expressed by both the inverse product (Lemma 7.17.2) and the cross ratio. Much of the hyperbolic geometry required for a detailed discussion of Fuchsian groups is best described in terms of geodesic pencils.
Prove that the three perpendicular bisectors of the sides of a hyperbolic triangle lie in one pencil. In fact, any such isometry can be expressed as a product of two involutions, and the geometric operation of isometry is closely related to the theory of pencils. The g axis (in the hyperbolic plane) is the pencil axis, that is, a unique geodesic length that is perpendicular to all lines in the pencil and ends at fixed points g.
Finally, note that in all cases the level curves of the displacement function are exactly the curves in the family perpendicular to the pencil associated with g. It is this geometric proof that reveals the true nature of the isometric circle in plane hyperbolic geometry. We know that any conformal isometry of the hyperbolic plane can be expressed as a product f = a2 of reflections (Ti in geodesics L1.
We may assume that the unit disk A (or the half-plane H2) is G-invariant and thus we can regard G as a discrete group of isometries of the hyperbolic plane. If G is non-elementary then (Theorem 5.3.7) the limit set A of G lies on the unit circle (this also applies to elementary Fuchsian groups) and it is important to distinguish between the cases where A is or is not the whole circle. We say that G is of the first kind if A = ÔDen of the second kind if A is a real subset of.
If g is elliptic, it is primitive when generating the stabilizer and has a rotation angle of the form 2 it/n. In some, but not all, cases this can be described in terms of a trace function.
