By Iain Adamson
This ebook has been known as a Workbook to make it transparent from the beginning that it's not a traditional textbook. traditional textbooks continue by means of giving in every one part or bankruptcy first the definitions of the phrases for use, the innovations they're to paintings with, then a few theorems related to those phrases (complete with proofs) and at last a few examples and routines to check the readers' knowing of the definitions and the theorems. Readers of this publication will certainly locate the entire traditional constituents--definitions, theorems, proofs, examples and exercises yet no longer within the traditional association. within the first a part of the publication might be chanced on a brief assessment of the fundamental definitions of basic topology interspersed with a wide num ber of routines, a few of that are additionally defined as theorems. (The use of the be aware Theorem isn't meant as a sign of trouble yet of significance and value. ) The workouts are intentionally now not "graded"-after all of the difficulties we meet in mathematical "real life" don't are available order of hassle; a few of them are extremely simple illustrative examples; others are within the nature of educational difficulties for a conven tional path, whereas others are rather tricky effects. No ideas of the routines, no proofs of the theorems are integrated within the first a part of the book-this is a Workbook and readers are invited to attempt their hand at fixing the issues and proving the theorems for themselves.
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Proof. I5, we have 'rhus we need only establish the second inequality in the statement of the Theorem, and we may assume p E K . 14 for the constants (Ro, R 1). Let N be the largest integer such that d(p, f:) K) ~ N r . 14 N times to conclude that Mv(P) ~ 1 2N sup Mv(q). 15, so the stated bound on Mv(p) follows. II To give a global version of the preceding result, we need to show a point deep in the convex core remains reasonably deep after a (Illasi-isometry. 16 Let ~ : JHrl ~ JHrl be an L-quasi-isometry, and let A be a closed subset of S~-l.
Finally suppose '¢ is reducible and of infinite order; to complete the proof, we will show Qoo has an accidental parabolic. For simplicity, assume there is a single simple nonperipheralloop 9 on S preserved by 'l/J up to isotopy (the case of a system of loops is similar). Pass to a subsequence such that Qi ~ Qoo algebraically. Then the hyperbolic length of the geodesic representative of 9 in Qi converges to a limit L. If L = 0 then 9 is an accidental parabolic in Qoo as desired. Now suppose L > o.
We will work in the universal cover, normalizing so p = O. If the Lemma is false we can find a sequence of Kleinian groups ri, a sequence of deformations Vi and a sequence of radii ri ---. 00, such that MVi(P) > 1/2 but MVi(q) $ 1 on the sphere S(p, ri) C Ki. 18). Since the convergence is uniform, Mv(P) = limMvi(P) ~ 1/2. But since M is rigid, v is conformal, and therefore Mv = O. The Lemma follows· by contradiction. • Next we show the visual extension of a deformation tends to an exponentially fast in the convex core.