The second edition is a thorough revision of the first edition that embodies
hundreds of changes, corrections, and additions, including over sixty new
lemmas, theorems, and corollaries. The following theorems are new in the
second edition: 1.4.1, 3.1.1, 4.7.3, 6.3.14, 6.5.14, 6.5.15, 6.7.3, 7.2.2, 7.2.3,
7.2.4, 7.3.1, 7.4.1, 7.4.2, 10.4.1, 10.4.2, 10.4.5, 10.5.3, 11.3.1, 11.3.2, 11.3.3,
11.3.4, 11.5.1, 11.5.2, 11.5.3, 11.5.4, 11.5.5, 12.1.4, 12.1.5, 12.2.6, 12.3.5,
12.5.5, 12.7.8, 13.2.6, 13.4.1. It is important to note that the numbering
of lemmas, theorems, corollaries, formulas, figures, examples, and exercises
may have changed from the numbering in the first edition.
The following are the major changes in the second edition. Section 6.3,
Convex Polyhedra, of the first edition has been reorganized into two sections, §6.3, Convex Polyhedra, and §6.4, Geometry of Convex Polyhedra.
Section 6.5, Polytopes, has been enlarged with a more thorough discussion
of regular polytopes. Section 7.2, Simplex Reflection Groups, has been
expanded to give a complete classification of the Gram matrices of spherical, Euclidean, and hyperbolic n-simplices. Section 7.4, The Volume of a
Simplex, is a new section in which a derivation of Schl¨afli’s differential formula is presented. Section 10.4, Hyperbolic Volume, has been expanded to
include the computation of the volume of a compact orthotetrahedron. Section 11.3, The Gauss-Bonnet Theorem, is a new section in which a proof
of the n-dimensional Gauss-Bonnet theorem is presented. Section 11.5,
Differential Forms, is a new section in which the volume form of a closed
orientable hyperbolic space-form is derived. Section 12.1, Limit Sets of Discrete Groups, of the first edition has been enhanced and subdivided into
two sections, §12.1, Limit Sets, and §12.2, Limit Sets of Discrete Groups.
The exercises have been thoroughly reworked, pruned, and upgraded.
There are over a hundred new exercises. Solutions to all the exercises in
the second edition will be made available in a solution manual.
Finally, I wish to express my gratitude to everyone that sent me corrections and suggestions for improvements. I especially wish to thank Keith
Conrad, Hans-Christoph Im Hof, Peter Landweber, Tim Marshall, Mark
Meyerson, Igor Mineyev, and Kim Ruane for their suggestions.