# 2-Signalizers of finite groups - download pdf or read online

By Mazurov V. D.

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Then there is a sequence of distinct elements of G converging to the identity. Proof. Since G is not discrete, there is a sequence of elements { gm } of G converging to some element g e M. Normalize G so that g is either z - z + 1, or of the form z - k2z. t converges to the identity. 3. Proposition. Let G be a Kleinian group. Then G is a discrete subgroup of M. 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.

20. Let D be a fundamental domain for the Kleinian group G. Suppose there is a connected component Do of D so that every side of Do is paired with another (or the same) side of Do, and so that these side pairing transformations generate G. , g(A0) = Ao for all g e G). L. I. The usual terminology for what we call "free discontinuity" is either "discontinuity" or "proper discontinuity", but the terminology is not standard. 5. The precise definition of "Kleinian group" has varied with time. The definition used here is a bit old-fashioned.

3. Corollary. Let p: (9, zo) -+ (X, xo) be a regular covering, and let w be a loop on X based at some point x. Then w lifts to a loop starting at some point in the fiber over x if and only if w lifts to a loop starting at any other point in the fiber over x. 4. The converse to the above is also true. Proposition. Let p: ()7, go) - (X, xo) be a covering. If every loop based at xo, which lifts to a loop starting at go, also lifts to a loop starting at any other point x` in the fiber over xo, then the covering is regular.