Albert Marden, one of the principal architects of topological research related to Kleinian groups since the last quarter of the 20th century, has written a “modern classic” on the topology and geometry of two and three dimensional hyperbolic manifolds. Marden’s book is a new and revised edition, with a fresh title and a significant amount of updated material, of his 2007 text *Outer Circles: An Introduction to Hyperbolic 3-Manifolds*.

His unusual, though altogether excellent, text amply demonstrates the “joie de vivre” experienced by a specialist research mathematician via the medium of an exciting graduate seminar. In Marden’s words:

My idea was to try to make the subject accessible to beginning graduate students with minimal specific prerequisites. Yet I wanted to leave students with more than a routine compendium of elementary facts. Rather I thought students should see the big picture, as if climbing a watchtower to overlook the forest. Each student should end his or her studies having a personal response to the timeless question: What is this good for?

With such thoughts in mind, I have tried to give a solid introduction and at the same time to provide a broad overview of the subject as it is today.

The first six chapters form the core of the text. Of these the first two, “Hyperbolic space and its isometries” and “Discrete groups”, provide more details of proofs than the rest. The next two chapters, “Properties of hyperbolic manifolds” and “Algebraic and geometric convergence”, move faster, with readers pointed to the references for details. The fifth and sixth chapters, “Deformation spaces and the ends of manifolds” and “Hyperbolization”, are largely expository; they survey a wide sweep of recent research with almost no proofs. The final two chapters, “Line geometry” and “Right hexagons and hyperbolic trigonometry”, are not quite connected to the main thread of the first six, and appear almost as an appendix where Marden exposes an approach to various classical formulae from hyperbolic trigonometry due to his collaborator Troels Jørgensen.

Hyperbolic surfaces or 3-manifolds are quotients of 2- or 3-dimensional hyperbolic (constant negative curvature) space by torsion-free discrete subgroups (known as *Fuchsian* or *Kleinian* groups) of their respective isometry groups. Poincaré, Fricke, and Klein were among the earliest explorers of the great expanse of mathematics that related to the study of such objects, from the time of their birth in the late 19th century. The diagram on the left, taken from Cannon-Floyd-Kenyon-Parry’s excellent introduction to Hyperbolic Geometry in *Flavors of Geometry* (MSRI Pub. Vol. 31, 59-115), gives the reader a bird’s eye view of this rich terrain.

The major epochs in the modern study of Kleinian groups include the the injection of analytic methods by Ahlfors and Bers in the 1960s, Thurston’s topological revolution in the 1970s, and then the dynamical systems and geometric group theory viewpoints injected by Sullivan and Gromov in the 1980s. The area witnessed a flurry of activity in the last couple of decades: first with the resolutions of three major conjectures (now theorems) — the tameness conjecture, the density conjecture, and the ending lamination conjecture — in the early 2000s, followed by further breakthrough results by Jeremy Kahn, Vladimir Markovic, Ian Agol, Daniel Wise and Mahan Mj in the last couple decades. Marden’s text is currently one of the only texts aimed at graduate students that surveys a large swath of all this rather recent activity.

The book’s core, which makes it stand apart from now-standard textbook treatments of hyperbolic geometry/manifolds like those by Benedetti-Petronio (*Lectures on Hyperbolic Geometry*, Springer, 1992) and Ratcliffe, is to be found in the incredible array of “Exercises and Explorations” that end each chapter, and that often constitute over half of each! This is perhaps where Marden’s voice “in seminar mode” may be discerned most clearly. Certain students encountering this wealth of material may feel overwhelmed, especially without some guidance near at hand. I predict, however, that there will also be those who will be moved to discover exciting research directions (and potentially their thesis work!) hidden between these lines.

The diagrams, over 60 in number and for the most part highly intricate computer-generated graphics, will leave the reader craving for more. For a taste, marvel at the gorgeous frontispiece — *Thurston’s Jewel*, created by Jeffrey Brock and David Dumas. Sadly, the author makes no attempt to lead the interested reader to further resources about such intriguing figures. It would be fruitful to have a supplemental website devoted to archiving documentation and the programs that create such images. I also felt that the text would greatly benefit from the inclusion of even more diagrams and figures to help elucidate various aspects of the (potentially alarming!) profusion of concepts, theorem-statements, and proof-sketches. Even a healthy infusion of hand-drawn illustrations, as in Thurston’s seminal Princeton notes, would go a long way in making the book more accessible.

There are a few typos, mostly minor errors in spelling and mathematical grammar, but no more than what one would expect for a book of this size and such do not detract from the mathematical content at hand.

Any reviewer of a book that covers such vast ground will find instances of mathematical content that could easily have been added, but whose exclusion does not in any way diminish the book’s virtues. Here are three samples of such “potential extensions” regarding a particular topic. After introducing the fundamental concept of the *limit set* of a Kleinian group, Marden lists some of its most useful properties in a eight-part lemma. However, the key property — the *minimality* of the limit set — is missing. The limit set is the smallest (in the sense of inclusion) group invariant compact subset of the boundary of the appropriate hyperbolic space on which the group acts. The end-of-chapter exploration “No tangents at loxodromic fixed points” sketches a proof of the beautiful fact that the limit set cannot have a tangent line at a fixed point of a loxodromic element of the group unless the limit set is Möbius equivalent to a circle. This is due originally to Fricke (see his “Die Kreisbogenvierseite und das Princip der Symmetrie” (*Math. Ann.* 44 (4) (1894) 565–599) in response to a conjecture of Poincaré. There is an amusing history that surrounds this result that the interested reader may find in the second footnote of Das-Simmons-Urbański’s “Dimension rigidity in conformal structures” (*Adv. in Math.*, Vol. 308, (2017), 1127–1186). In the same vein, the end-of-chapter explorations on the modular group \(\mathrm{SL}(2,\mathbb{Z})\) and the Farey sequence could be followed up with one on Diophantine approximation within the limit set of a Kleinian group, see e.g., Paulin’s survey “A survey of some arithmetic applications of ergodic theory in negative curvature”.

Much of the mathematics Marden exposes is classical and inhabited by concrete low-dimensional formulae and closed-form expressions that could appear as reflections of the deep-rooted complex-analytic heritage of the subject. For younger students interested in seeing what extends beyond the constant curvature setting, the study of CAT(\(\kappa\)) spaces and Gromov hyperbolicity beckons. Bridson-Haefliger’s *Metric spaces of non-positive curvature* (Springer 1999) is an almost canonical reference today, and has a view towards applications in geometric group theory. For those interested in dynamics, see Quint’s lucid *Overview of Patterson-Sullivan theory*, Roblin’s SMF memoir *Ergodicité et équidistribution en courbure négative* and Paulin-Pollicott-Shapira’s recent *Astérisque* volume *Equilibrium states in negative curvature*.

Marden closes his preface with the following words:

So here we are today, nearly 130 years after Poincaré and approaching 200 after the initial ferment of ideas of Gauss. We are witnessing a full flowering of the vision and struggle for understanding of the nineteenth-century masters. Still, the final word remains an elusive goal.

Echoing Marden’s sentiment, I cannot resist ending this review without a signpost for future explorers of the brave new worlds that higher-dimensional Kleinian groups inhabit. One could do worse than begin with Kapovich’s excellent survey *Kleinian Groups in Higher Dimensions* (Progress in Mathematics, Vol. 265, 485-562, Birkhäuser 2007).

Tushar Das is an Associate Professor of Mathematics at the University of Wisconsin–La Crosse.