By Randy Jolly

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Over the last 30 years the idea of finite teams has built dramatically. Our realizing of finite uncomplicated teams has been more advantageous by means of their type. many questions about arbitrary teams might be lowered to related questions on uncomplicated teams and functions of the speculation are starting to seem in different branches of arithmetic.

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Proof. Suppose not. Then there is a sequence { gm } of distinct elements of G, with gm - 1; so gm(z) -+ z for every z. Hence for every point z e C, either there are infinitely many of the gm with gm(z) = z, or there are infinitely many translates of z in every neighborhood of z. In either case, z # °Q. 4. The converse to the above is false. G. S. The proposition below is sometimes known in the literature as the ShimizuLeutbecher lemma. Proposition. Let G be a discrete subgroup of ADO, where G contains f(z) = z + 1.

If Y,, .... Y are disjoint connected subsets of Q/G, and if T. is a connected component of p-' (Ym), then not only is each T. a panel, but for k # m, g(Tk) fl T. = 0 for all Y E G. In general, we say that (T, . . is precisely invariant under (J...... in G, if each T. is precisely invariant under Jm, and if for m : k, and for all g e G, g(Tm)fl Tk = Q. J. 1. Let cp: G - G* be an isomorphism between Kleinian groups. Since an element of G has finite order if and only if it is elliptic, qp preserves elliptic elements.

Arcs, paths, and loops) of (iii) Every element of F meets every other element of F in at most finitely many points. (iv) Every element of F is either a simple loop, or a path where each endpoint is either a special point or a point of some other element of F, or a proper arc, where each endpoint either lies on the boundary of S, or is a special point, or is a point of some other element of F (An endpoint of a proper arc on the boundary of S need not be well defined. ) If every connected component of S - F is simply connected, then the dissection is full.