defined by the equation x′ = ax + b, where a and b are any rational
numbers, is improperly discontinuous; and the group defined by
x′ = x + a, where a is an integer, is properly discontinuous, whatever
the range of the variable. On the other hand, the group, to be later
considered, defined by the equation x′ = ax + bcx + d, where a, b, c, d are
integers satisfying the relation ad − bc = 1, is properly discontinuous
when x may take any complex value, and improperly discontinuous
when the range of x is limited to real values.
Among the discontinuous groups that occur in analysis, a large number may be regarded as arising by imposing limitations on the range of variation of the parameters of continuous groups. If
x′s = ƒs (x1, x2, . . ., xn; a1, a2, . . ., ar), (s = 1, 2, . . ., n),
are the finite equations of a continuous group, and if C with parameters c1, c2, . . ., cr is the operation which results from carrying out A and B with corresponding parameters in succession, then the c’s are determined uniquely by the a’s and the b’s. If the c’s are rational functions of the a’s and b’s, and if the a’s and b’s are arbitrary rational numbers of a given corpus (see Number), the c’s will be rational numbers of the same corpus. If the c’s are rational integral functions of the a’s and b’s, and the latter are arbitrarily chosen integers of a corpus, then the c’s are integers of the same corpus. Hence in the first case the above equations, when the a’s are limited to be rational numbers of a given corpus, will define a discontinuous group; and in the second case they will define such a group when Linear discontinuous groups. the a’s are further limited to be integers of the corpus. A most important class of discontinuous groups are those that arise in this way from the general linear continuous group in a given set of variables. For n variables the finite equations of this continuous group are
x′s = as1x1 + as2x2 + . . . + asnxn, (s = 1, 2, . . ., n),
where the determinant of the a’s must not be zero. In this case the c’s are clearly integral lineo-linear functions of the a’s and b’s. Moreover, the determinant of the c’s is the product of the determinant of the a’s and the determinant of the b’s. Hence equations (ii.), where the parameters are restricted to be integers of a given corpus, define a discontinuous group; and if the determinant of the coefficients is limited to the value unity, they define a discontinuous group which is a (self-conjugate) subgroup of the previous one.
The simplest case which thus presents itself is that in which there are two variables while the coefficients are rational integers. This is the group defined by the equations
| x′ = ax + by, | |
| y ′ = cx + dy, |
where a, b, c, d are integers such that ad − bc = 1. To every operation of this group there corresponds an operation of the set defined by
z′ = az + bcz + d ,
in such a way that to the product of two operations of the group there corresponds the product of the two analogous operations of the set. The operations of the set (iv.), where ad − bc = 1, therefore constitute a group which is isomorphic with the previous group. The isomorphism is multiple, since to a single operation of the second set there correspond the two operations of the first for which a, b, c, d and −a, −b, −c, −d are parameters. These two groups, which are of fundamental importance in the theory of quadratic forms and in the theory of modular functions, have been the object of very many investigations.
Another large class of discontinuous groups, which have far-reaching
applications in analysis, are those which arise in the first
instance from purely geometrical considerations. By the
combination and repetition of a finite number of geometrical
operations such as displacements, projective
Discontinuous groups arising
from geometrical operations.transformations, inversions, &c., a discontinuous group of
such operations will arise. Such a group, as regards the
points of the plane (or of space), will in general be improperly
discontinuous; but when the generating operations
are suitably chosen, the group may be properly
discontinuous. In the latter case the group may be
represented in a graphical form by the division of the plane (or space)
into regions such that no point of one region can be transformed into
another point of the same region by any operation of the group,
while any given region can be transformed into any other by a
suitable transformation. Thus, let ABC be a triangle bounded by
three circular arcs BC, CA, AB; and consider the figure produced
from ABC by inversions in the three circles of which BC, CA, AB are
part. By inversion at BC, ABC becomes an equiangular triangle
A′BC. An inversion in AB changes ABC and A′BC into equiangular
triangles ABC′ and A″BC′. Successive inversions at AB and BC
then will change ABC into a series of equiangular triangles with B
for a common vertex. These will not overlap and will just fill in the
space round B if the angle ABC is a submultiple of two right angles.
If then the angles of ABC are submultiples of two right angles (or
zero), the triangles formed by any number of inversions will never
overlap, and to each operation consisting of a definite series of
inversions at BC, CA and AB will correspond a distinct triangle into
which ABC is changed by the operation. The network of triangles so
formed gives a graphical representation of the group that arises from
the three inversions in BC, CA, AB. The triangles may be divided
into two sets, those, namely, like A″BC′, which are derived from ABC
by an even number of inversions, and those like A′BC or ABC′ produced
by an odd number. Each set are interchanged among themselves
by any even number of inversions. Hence the operations
consisting of an even number of inversions form a group by themselves.
For this group the quadrilateral formed by ABC and A′BC constitutes
a region, which is changed by every operation of the group into
a distinct region (formed of two adjacent triangles), and these regions
clearly do not overlap. Their distribution presents in a graphical
form the group that arises by pairs of inversions at BC, CA, AB; and
this group is generated by the operation which consists of successive
inversions at AB, BC and that which consists of successive inversions
at BC, CA. The group defined thus geometrically may be presented
in many analytical forms. If x, y and x′, y ′ are the rectangular co-ordinates
of two points which are inverse to each other with respect
to a given circle, x′ and y ′ are rational functions of x and y, and conversely.
Thus the group may be presented in a form in which each
operation gives a birational transformation of two variables. If
x + iy = z, x′ + iy ′ = z′, and if x′, y ′ is the point to which x, y is transformed
by any even number of inversions, then z′ and z are connected
by a linear relation z′ = αz + βγz + δ, where α, β, γ, δ are constants (in
general complex) depending on the circles at which the inversions are
taken. Hence the group may be presented in the form of a group
of linear transformations of a single variable generated by the two
linear transformations z′ = α1z + β1γ1z + δ1, z′ = α2z + β2γ2z + δ2, which correspond
to pairs of inversions at AB, BC and BC, CA respectively. In
particular, if the sides of the triangle are taken to be x = 0, x2 + y 2 −
1 = 0, x2 + y 2 + 2x = 0, the generating operations are found to be
z′ = z + 1, z′ = −z−1; and the group is that consisting of all transformations
of the form z′ = az + bcz + d, where ad − bc = 1, a, b, c, d being
integers. This is the group already mentioned which underlies the
theory of the elliptic modular functions; a modular function being
a function of z which is invariant for some subgroup of finite index of
the group in question.
The triangle ABC from which the above geometrical construction started may be replaced by a polygon whose sides are circles. If each angle is a submultiple of two right angles or zero, the construction is still effective to give a set of non-overlapping regions, which represent graphically the group which arises from pairs of inversions in the sides of the polygon. In their analytical form, as groups of linear transformations of a single variable, the groups are those on which the theory of automorphic functions depends. A similar construction in space, the polygons bounded by circular arcs being replaced by polyhedra bounded by spherical faces, has been used by F. Klein and Fricke to give a geometrical representation for groups which are improperly discontinuous when represented as groups of the plane.
The special classes of discontinuous groups that have been dealt with in the previous paragraphs arise directly from geometrical considerations. As a final example we shall refer briefly to a class of groups whose origin is essentially analytical. LetGroup of a linear differential equation.
| d ny | + P1 | d n−1y | + . . . + Pn−1 | dy | + Pny = 0 |
| dxn | dxn−1 | dx |
be a linear differential equation, the coefficients in which are rational functions of x, and let y1, y2, . . ., yn be a linearly independent set of integrals of the equation. In the neighbourhood of a finite value x0 of x, which is not a singularity of any of the coefficients in the equation, these integrals are ordinary power-series in x − x0. If the analytical continuations of y1, y2, . . ., yn be formed for any closed path starting from and returning to x0, the final values arrived at when x0 is again reached will be another set of linearly independent integrals. When the closed path contains no singular point of the coefficients of the differential equation, the new set of integrals is identical with the original set. If, however, the closed path encloses one or more singular points, this will not in general be the case. Let y ′1, y ′2, . . ., y ′n be the new integrals arrived at. Since in the neighbourhood of x0 every integral can be represented linearly in terms of y1, y2, . . ., yn, there must be a system of equations
| y ′1 = a11y1 + a12y2 + . . . + a1nyn, |
where the a’s are constants, expressing the new integrals in terms of the original ones. To each closed path described by x0 there therefore corresponds a definite linear substitution performed on the y’s. Further, if S1 and S2 are the substitutions that correspond to two closed paths L1 and L2, then to any closed path which can be continuously deformed, without crossing a singular point, into L1 followed by L2, there corresponds the substitution S1S2. Let L1, L2, . . ., Lr be arbitrarily chosen closed paths starting from and returning to the same point, and each of them enclosing a single one of the
