A crash course on Kleinian groups; lectures given at a by American Mathematical Society

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Additional info for A crash course on Kleinian groups; lectures given at a special session at the January 1974 meeting of the American Mathematical Society at San Francisco

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Where convenient, we will drop the base points, and simply refer to a regular covering p: 2 -- X, with defining (normal) subgroup N. 6. Theorem. Let p: )Z -.

7. Ahlfors' finiteness theorem appeared in [4]; a proof can be found in Kra [43 pg. 333-338]. Chapter III. Covering Spaces This chapter is primarily a review of standard covering space theory, including some easy, but not so will known facts about regular coverings. There is also a discussion of branched regular coverings, in dimension two, and a proof of the branched universal covering surface theorem. Throughout this chapter, all spaces are connected manifolds of some fixed dimension, which, for most of our purposes, can be taken to be two.

D. The Limit Set 21 As we saw above, we can have am+, = 0 if and only if dm+t = 0. 4), this can occur only if k° = 1; since f is loxodromic, this cannot occur. Hence am+, and dm+, are both 0. 4), and the induction hypothesis. Since I bmcml - 0, we can extract a subsequence { g, } of distinct elements of G. S) am+t =(I +bmcm)k-bmcmk-' =k+bmcm(k-k-) 4k, dm+1 = (I + bmcm)k-t - bmCmk = k-' + bmcm(k-' - k) - k'. If we knew that bm and cm were bounded, we could extract a convergent subsequence. The next lemma asserts that for m sufficiently large, we can conjugate gm by some power of f so as to achieve this result.

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