On the Formation of Groups of Linear Transformations by Combination

VOL. 13, 1927

MA THEMA TICS: L. R. FORD

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formations Si, S2, .... constitute a set of generating transformations for the group; otherwise they do not. 1 NATIONAL RESEARCH FZLLOW IN MATHZMATICS. Certain results of this note in the special case of Fuchsian groups have been published. L. R. Ford, "The Fundamental Region for a Fuchsian Group," Bull. Amer. 2

Math. Soc., 31, 531-539 (1925).

ON THE FORMATION OF GROUPS OF LINEAR TRANSFORMATIONS BY COMBINATION By LZSTZR R. FoRDi LIuPZIG, GERMANY Communicated March 15, 1927

In this note we discuss groups formed by combining a finite or infinite number of known groups. By suitable combinations we can set up a great variety of Kleinian groups. Our present knowledge of Kleinian groups is not extensive. These groups in all their generality present a complexity and a richness which are largely unfathomed. By the method here described we can form broad classes of groups which bring out certain of the intricate possibilities. In a previous note we gave a method of constructing a fundamental region for a group.2 The fundamental region consists of all those points z such that in the neighborhood of z there are no points interior to an isometric circle of a transformation of the group. We consider a set of grotps ri, r2, .... finite or infinite in number, such that the isometric circles of the transformations of each group are exterior to (external tangency will be permitted) the isometric circles of the transformations of all the other groups. Let the transformations of rF, r2, . be combined in all possible ways to form a group r. We have two kinds of transformations: (1) Those belonging to the original groups; and (2) cross products resulting from the combination of transformations not all belonging to the same original group. Our major result is the following: THZORZM.-The isometric circle of each cross product lies within the isometric circle of some transformation belonging to one of the original groups. It follows from this theorem that in the formation of the fundamental region for r we can disregard the isometric circles of all cross products. The fundamental region R for r consists of all points common to the fundamental regions R1, R2, . of ri, r2, ...., exclusive of those common points (if any) which have an infinite number of boundary points of R1, R2. ...., in their neighborhoods.

MA THEMA TICS: L. R. FORD

290-

PROC. N. A. S.

The proof of the theorem depends upon the following lemma. We represent the isometric circle of a transformation T by It and the isometric circle of its universe by I. We use the symbol < to mean "is contained within." LzMMA.-If I and I are exterior to one another then for the transformation U = TS we have I. < I, The proof is simple. Let z be a point outside (or on) Is. Then S carries z into z' within (or on) 1 with decrease of lengths (or without alteration of lengths) in the neighborhood3 of z. Now z' is outside It; so T transforms z' with decrease of lengths. Hence U transforms z with decrease of lengths, whence z is outside I,. Since every point on or outside I, is outside I,., the latter circle is contained in the former. Let T('), T(). be the transformations of ri. As a convenient notation let the subscripts be so assigned that T(),, is the inverse of TIPm. [There are thus two symbols for a self-inverse transformation, but this leads to no confusion. ] The group r consists of all transformations of the form T- T("k) mk

) T("kmk-l

m2

fl

We can suppose that no factor is the identical transformation, since it can be cancelled; also that no two successive indices ni, ni + 1 are the same, for the two transformations can be combined into a single transformation of the group rF. We shall represent the isometric circle of T by I(k,... . m) We shall prove that ( (nk, nk-1 * n < I(nk_1. ...Inl) nl) < I(ni) I(M2, < ...< (a *** ml .**, M2, mI mk, mk-1,

,ml

Mk-1,

ml

The inequality holds for all combinations of two transformations I(n2, ni) m2, ml