The purpose of Outer Circles We live in a three-dimensional space; what sort of space is it? The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis.

The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations.

## Outer circles: An introduction to hyperbolic 3-Manifolds — [email protected]

There is an extensive bibliography for further study. Get A Copy.

- Outer Circles: An Introduction To Hyperbolic 3 Manifolds ( Cambridge University Press )!
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Community Reviews. Showing Rating details. More filters. Sort order. Aug 02, Alun Williams rated it liked it. I bought this book hoping to deepen my knowledge of Kleinian Groups and Hyperbolic Geometry and the connections between them, having been introduced to the first topic through the wonderful "Indra's Pearls", and after being baffled by Thurston's book on geometry and topology. I found this book fascinating and frustrating in almost equal measure, with some chapters the more geometric ones proving far more comprehensible to me than the chapters actually focussing on manifolds.

For me the best part I bought this book hoping to deepen my knowledge of Kleinian Groups and Hyperbolic Geometry and the connections between them, having been introduced to the first topic through the wonderful "Indra's Pearls", and after being baffled by Thurston's book on geometry and topology. For me the best part of the book are the "exercises and explorations" at the end of the chapters, almost all of which are very interesting, and which I thought most revealed the depth of the author's knowledge of his field.

### An introduction to hyperbolic 3-Manifolds

The book is also beautifully illustrated, although this was also one of my frustrations: the connection between the illustrations quite a few of which I have seen elsewhere , and the text, is not always at all clear, and I would have very much liked to learn more about some of the illustrations were produced. My main problems with the book, other than coming to the conclusion that I will never understand hyperbolic manifolds well : 1 Some of the proofs are very unsatisfying, relying on results that are only proved later in the book, or in some cases not at all.

Sep 13, Steve Dalton rated it it was ok Shelves: math. This book is full of facts about hyperbolic 3 manifolds with absolutely no connecting thread between these facts. There is virtually no motivation, big picture, "here's where we are going" in almost pages.

## Limits of limit sets I

I like this subject a lot, but I just don't see any organization in this book. At all. Beautiful figures adorn some pages, with nary a reference to them in the main text. The exercises are really just eve This book is full of facts about hyperbolic 3 manifolds with absolutely no connecting thread between these facts.

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The exercises are really just even more loosely connected ideas, distilled from research papers. There is no "exercise" here: you won't be solving anything. Sure the book has big connecting ideas on the global level: hyperbolization, kleinian groups, etc. But locally, I found myself constantly asking "who cares about pleated surfaces? But even the bibliography is lacking: where is Benson and Farb's book or the Bridson-Haefliger classic?

Apollonian circles, hyperbolic geometry Mathematics Subject Classification Bolyai and N. Lobaschevsky [Mi]. Its discovery solved the two millenia old problem about the role of the Parallel Postulate in Euclidean geometry: this postulate cannot be left out from the Euclidean geometry. Hyperbolic geometry provides an example of a geometry which satisfies all the postulates of the Euclidean geometry except that the Par- allel Postulate does not hold.

During the past two decades the hyperbolic geometry has surfaced in a number of contexts which do not belong to ge- ometry proper: geometric function theory, discrete group theory, modern theory of quasiconformal and quasiregular mappings in n-dimensional Eu- clidean spaces and also in fields such as relativity theory and graphical art of Escher [SC]. In function theoretic applications the hyperbolic metric is often more natural than the Euclidean metric. All these developments have lead to attempts to generalize hyperbolic metric to subdomains of the Euclidean space.

For instance the quasihyperbolic metric of F. Gehring and B. Palka [GP] and the Apollonian metric originally introduced by D. Barbilian [Ba] and rediscovered by A. For an interesting historical survey of the Apollonian metric see [BS].

## Albert Marden

Our goal is to keep the prerequisites for reading this paper as minimal as possible and, this in mind, we try to list carefully the necessary basic information in the introduction. We often identify R2 with the complex plane C. This means that Apollonian balls with fixed base points are ordered by inclusion. The following lemma gives the Euclidean center point and the radius of an Apollonian circle.

Observe that the absolute ratio depends on the order of points and e. The Apollonian distance defines a metric, whenever the complement of the domain G is not contained in a hyperplane [Be2]. By [Be1, p. With the respective weight functions given above the definitions 1. In particular, 1. Hyperbolic geodesics are arcs of circles that are orthogonal to the bound- ary of the domain.

More precisely, a hyperbolic geodesic segment is the intersection of the domain with the Euclidean circle or straight line which is orthogonal to the boundary of the domain, see [Be1]. For any two distinct points the hyperbolic geodesic segment is unique.

The above basic facts can be found in our standard references [Be1, A2] and in many other sources on hyperbolic geometry such as [A, BM, KL].

Farreaching and specialized advanced texts discussing hyperbolic geometry include [R, Mar]. Our plan here is to show the many links between Apollonian circles and hyperbolic geometry. Earle and L. Harris [EH].

Then it follows from 1. For that purpose the reader might wish to see [Kr, p. The next result gives a formula for hyperbolic geodesic segment in B2. Proof of Lemma 1. Now C is a circle with center point on the line l that contains x and y. We prove next orthogonality to S 1 0, 1. Finally, we prove orthogonality of S 1 w, r and C. Let us denote inter- section of S 1 w, r and C in B2 by z.

By Corollary 2. Therefore by Lemma 2. Lemma 2. We will now describe a geometric construction for finding the hyperbolic midpoint. The hyperbolic geodesic joining x and y is an arc of a circle S 1 a, r , where a and r depend only on x and y. Formulas for a and r are given in Lemma 2. Construct the circle S 1 a, r , which contains the hyperbolic geodesic joining x and y.

This is a well-known formula, see e.

By [V, 2. In Theorem 3. Let us denote the better lower bound of Theorem 3. We prove first the inequality. Let z be a point on the same geodesic with x and y. Thus by Lemma 2. This is a straightforward computation, which we omit. The following result gives the hyperbolic midpoint of two points x and y.

The next result is based on the following idea. Since 5.