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By John Hempel

It sort of feels unusual that no-one has reviewed this booklet formerly, yet unquestionably this can be a result of repute of the booklet and the truth that nearly all of these trying to find it don't have any want for a overview; in spite of the fact that, a minority might discover a assessment of what this e-book is and is not worthwhile of their choice to shop for or not.

What this e-book isn't really: 1) An advent to topology, or perhaps to low-dimensional topology. somebody who has heard of 3-manifolds and gotten excited could do higher to get a flavor of the topic in different places first, e.g. in Rolfsen's _Knots and Links_. 2) A learn monograph designed to deliver the reader up to the mark on present examine on 3-manifolds. This e-book is set 30 years previous and does not even point out the Geometrization Conjecture of Thurston. three) A ebook at the function of knot concept in 3-manifolds. Knots play a tremendous position within the thought, not just theoretically, yet as a wealthy resource of examples to sharpen the instinct and try out conjectures (through Dehn surgical procedures on knots and links). This position isn't mentioned during this book.

What this publication is: 1) A primer for topologists looking to turn into experts in 3-manifolds. the fundamental theorems relating to top decomposition, loop and sphere theorems, Haken hierarchy, and Waldhausen's theorems on Haken manifolds are defined intimately. those will be thought of a number of the highlights even if a lot correct fabric is unavoidably additionally defined. As possibly befitting a primer, the JSJ decomposition and attribute submanifold concept isn't integrated. Jaco's ebook enhances Hempel through overlaying this fabric. 2) A reference for these already acquainted with the fabric. The writing type is especially concise and to the purpose. This makes it basic to seem up a theorem to refresh one's reminiscence on a sticky element in an evidence. As an advent to the cloth, a few passages can be terse, yet necessarily after a few attempt, they are often "decoded" thoroughly, in contrast to a few texts that could be extra verbose yet can by no means be totally deciphered. i believe there can be a lot extra photographs; there will not be very many, to assert the least. but when the reader attracts his/her personal images, this just isn't an excessive amount of of a problem.

Some ultimate feedback: This e-book serves its twin position as a primer and reference admirably, however the reader might wander off within the info and lose the wooded area for the timber. regrettably, the one strategy to rectify this looks to learn a variety of papers at the topic to get an exceptional consider of a few of the threads that encourage present examine. yet with Hempel's _3-manifolds_ in hand, this job is far more uncomplicated and relaxing.

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Then it is clear that Γ is free of torsion and Γ w is compact for all w ∈ P1K (K) and LΓ = {0, ∞}. This example corresponds to Tate’s elliptic curve. 13. In the following let Γ be a Schottky group. For a γ = γA with c = 0 set √ |ad − bc| rγ := . |c| In the following we collect some properties of Schottky groups. 4. If Γ ⊂ PGL(2, K) is a Schottky group, then we have: (a) Γ is discrete. (b) Every γ ∈ Γ − {id} is hyperbolic. (c) If ∞ ∈ / LΓ , then for γ = γA ∈ Γ with A = γ = id ⇐⇒ ab cd γ (∞) = ∞ ∈ GL(2, K) we have: ⇐⇒ c = 0.

Follow. It remains to show that Γ acts discontinuously. We only have to explain that the closure of the orbit Γ w is compact for every w ∈ P1K (K). Since ∞ ∈ / LΓ , it suffices to show that every sequence (γi (w); i ∈ N) with pairwise distinct γi ∈ Γ admits a convergent subsequence. Put F (n) := γ (F ) = P1K − (Bn,1 ∪ · · · ∪ Bn,r(n) ), (γ )≤n where Bn,j are open discs contained in a large disc B0 := {z ∈ A1K ; |z| ≤ c} because of ∞ ∈ F . The symbol (γ ) indicates the number of elements used in a reduced representation of γ as a product of the αi .

It remains to show that E − ∩ γ (E) = ∅. For every point z ∈ E − we have |vγ (z)| < 1. Since 1 = vid (z) = vγ −1 γ (z) = vγ −1 γ (z) · vγ (z), / E for all γ ∈ Γ − {id}. it follows that |vγ −1 (γ (z))| > 1, and hence γ (z) ∈ (b) Let Γ := R(Γ ) ⊂ Γ be the subgroup generated by R(Γ ). Then Γ and Γ have the same fundamental domain, as follows from (a). Thus, P1K − α∈R(Γ ) Vα− is a fundamental domain for Γ as well. From (a) it follows ΩΓ ⊂ ΩΓ . Obviously we have that LΓ ⊂ LΓ , and hence ΩΓ = ΩΓ . Consider now some γ ∈ Γ .

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