By Lester R Ford

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Any region congruent to R o will serve as a region. Furthermore, we can replace any part of R o by a congruent part and still have a fundamental region. Thus, we can subtract a part at one boundary and add a congruent part at another. In this way the character of the bounding curve can be altered freely. -If no two points of a region are congruent, the transforms of the region by two distinct transformations of the group do not overlap. Let A be a region containing no two congruent points. Suppose that two transformations'of the group, Sand T, carry A into two overlapping regions.

If these networks fill up the whole plane, r t coincides with the original group; otherwise it does not. Let A be a closed region containing n6limit points of the group and having points in common with the region R. Then R and a SEC. 24] CYCLIC GROUPS 51 finite number of its transforms by the group r t cover A completely. For, along the sides of R which lie in A, we can adjoin regions of the network; along the sides of the new regions which lie in A, we can adjoin other regions of the network; and .

Let P lie on I t, and let P' on I t' be the point into which T carries P. We shall show that pI is also a boundary point of category ({3) •. We put aside the case pI = P, a situation which can arise only if It and It' coincide and P is a fixed point of the resulting elliptic transformation of period two; for this case has been included in (,,). First, pI is within no isometric circle. Suppose pI to be within Is; then S magnifies lengths in the neighborhood of P'. But, since T carries P into pI without alteration of lengths, ST magnifies lengths in the neighborhood of P.