,Foundations of Hyperbolic Manifolds
Third Edition
Solution Manual
John G. Ratcliffe
October 3, 2022
,Contents
1 Euclidean Geometry 1
1.1 Euclid’s Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Independence of the Parallel Postulate . . . . . . . . . . . . . . . 1
1.3 Euclidean n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Spherical Geometry 22
2.1 Spherical n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Elliptic n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Spherical Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Spherical Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . 36
3 Hyperbolic Geometry 42
3.1 Lorentzian n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Hyperbolic n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Hyperbolic Arc Length . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Hyperbolic Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . . . . 62
4 Inversive Geometry 71
4.1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Möbius Transformations . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Poincaré Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 The Conformal Ball Model . . . . . . . . . . . . . . . . . . . . . 85
4.6 The Upper Half-Space Model . . . . . . . . . . . . . . . . . . . . 89
4.7 Classification of Transformations . . . . . . . . . . . . . . . . . . 94
5 Isometries of Hyperbolic Space 107
5.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Groups of Isometries . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3 Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
i
, 5.4 Discrete Euclidean Groups . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Elementary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6 Geometry of Discrete Groups 137
6.1 The Projective Disk Model . . . . . . . . . . . . . . . . . . . . . 137
6.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.4 Geometry of Convex Polyhedra . . . . . . . . . . . . . . . . . . . 147
6.5 Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.6 Fundamental Domains . . . . . . . . . . . . . . . . . . . . . . . . 155
6.7 Convex Fundamental Polyhedra . . . . . . . . . . . . . . . . . . . 157
6.8 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 Classical Discrete Groups 166
7.1 Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2 Simplex Reflection Groups . . . . . . . . . . . . . . . . . . . . . . 169
7.3 Generalized Simplex Reflection Groups . . . . . . . . . . . . . . . 175
7.4 The Volume of a Simplex . . . . . . . . . . . . . . . . . . . . . . 178
7.5 Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . . 180
7.6 Torsion-Free Linear Groups . . . . . . . . . . . . . . . . . . . . . 182
8 Geometric Manifolds 185
8.1 Geometric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.2 Clifford-Klein Space-Forms . . . . . . . . . . . . . . . . . . . . . 188
8.3 (X, G)-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.4 Developing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9 Geometric Surfaces 202
9.1 Compact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.2 Gluing Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.3 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . 204
9.4 Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.5 Closed Euclidean Surfaces . . . . . . . . . . . . . . . . . . . . . . 212
9.6 Closed Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.7 Closed Hyperbolic Surfaces . . . . . . . . . . . . . . . . . . . . . 220
9.8 Hyperbolic Surfaces of Finite Area . . . . . . . . . . . . . . . . . 224
10 Hyperbolic 3-Manifolds 231
10.1 Gluing 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.2 Complete Gluing of 3-Manifolds . . . . . . . . . . . . . . . . . . . 236
10.3 Finite Volume Hyperbolic 3-Manifolds . . . . . . . . . . . . . . . 237
10.4 Hyperbolic Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10.5 Hyperbolic Dehn Surgery . . . . . . . . . . . . . . . . . . . . . . 248
ii
,11 Hyperbolic n-Manifolds 252
11.1 Gluing n-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 252
11.2 Poincaré’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 256
11.3 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . 258
11.4 Simplices of Maximum Volume . . . . . . . . . . . . . . . . . . . 260
11.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.6 Simplicial Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 271
11.7 Measure Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 274
11.8 Mostow Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
12 Geometrically Finite n-Manifolds 288
12.1 Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
12.2 Limit Sets of Discrete Groups . . . . . . . . . . . . . . . . . . . . 289
12.3 Limit Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
12.4 Geometrically Finite Discrete Groups . . . . . . . . . . . . . . . 298
12.5 Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
12.6 The Margulis Lemma . . . . . . . . . . . . . . . . . . . . . . . . 304
12.7 Geometrically Finite Manifolds . . . . . . . . . . . . . . . . . . . 311
12.8 Arithmetic Hyperbolic Groups . . . . . . . . . . . . . . . . . . . 315
13 Geometric Orbifolds 320
13.1 Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
13.2 (X, G)-Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
13.3 Developing Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 326
13.4 Gluing Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
13.5 Poincaré’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 336
iii
,Chapter 1
Euclidean Geometry
1.1 Euclid’s Parallel Postulate
1.2 Independence of the Parallel Postulate
Exercise 1.2.1 Let P be a point outside a line L in the projective disk model.
Show that there exists two lines L1 and L2 passing through P parallel to L such
that every line passing through P parallel to L lies between L1 and L2 . The two
lines L1 and L2 are called the parallels to L at P . All the other lines passing
through P parallel to L are called ultraparallels to L at P . Conclude that there
are infinitely many ultraparallels to L at P .
Solution: The lines L1 and L2 are the two lines passing through P that end
at the two ideal endpoints of L. See Figure 1.2.1. There are obviously infinitely
many lines in the projective disk model passing through P lying between L1
and L2 , and so there are infinitely many ultraparallels to L at P .
Exercise 1.2.2 Prove that any triangle in the conformal disk model, with a
vertex at the center of the model, has angle sum less than 180◦ .
Solution: Let ∆ be a triangle with vertices A, B, C in the conformal disk model
with C the center of the model. The sides AC and BC of ∆ are Euclidean line
segments. Let L be the hyperbolic line passing through the points A and B.
Then L is a circular arc that is orthogonal to the circle at infinity. The hyperbolic
half-plane bounded by L that does not contain ∆ is Euclidean convex, since it
is the intersection of two disks. Therefore, side AB of ∆ is a circular arc that
is contained in the corresponding Euclidean triangle ∆0 with vertices A, B, C.
The angle of ∆ at A is the angle between side AC and the Euclidean tangent
line T to the circular arc AB at A. The angle between line T and the Euclidean
line segment AB at A is positive. Hence, the angle of ∆ at A is less than the
angle of ∆0 at A. Likewise, the angle of ∆ at B is less than the angle of ∆0 at
1
,B. Therefore, the sum of the angles of ∆ is less than the sum of the angles of
∆0 , and so the sum of the angles of ∆ is less than 180◦ .
Exercise 1.2.3 Let u, v be distinct points of the upper half-plane model. Show
how to construct the hyperbolic line joining u and v with a Euclidean ruler and
compass.
Solution: If Re(u) = Re(v), draw the vertical ray L starting at Re(u) and
passing through u and v. Otherwise, draw the Euclidean line segment S joining
the points u and v. Construct the Euclidean perpendicular bisector M of the
line segment S. Let w be the point of intersection of M and the real axis. Then
|u − w| = |v − w|. Draw the semi-circle L centered at w passing through the
points u and v with endpoints on the real axis. Then in either case L is the
hyperbolic line joining u to v.
Exercise 1.2.4 Let φ(z) = az+b
cz+d with a, b, c, d in R and ad − bc > 0. Prove that
φ maps the complex upper half-plane bijectively onto itself.
Solution: Observe that
az + b cz + d ac|z|2 + adz + bcz + bd
· =
cz + d cz + d |cz + d|2
2
ac|z| + (ad + bc)Re(z) + (ad − bc)Im(z)i + bd
=
|cz + d|2
Hence, we have
az + b (ad − bc)Im(z)
Im = > 0.
cz + d |cz + d|2
Therefore φ maps the complex upper half-plane into itself.
Define real numbers a0 , b0 , c0 , d0 by the matrix equation
0 −1
a b0
a b
= .
c0 d0 c d
Observe that
a0 z+b0 aa0 z+ab0 bc0 z+bd0
a c0 z+d0 +b c0 z+d0 + c0 z+d0
=
a0 z+b0 ca0 z+cb0 dc0 z+dd0
c c0 z+d0 +d c0 z+d0 + c0 z+d0
(aa0 + bc0 )z + (ab0 + bd0 )
=
(ca0 + dc0 )z + (cb0 + dd0 )
= z,
since
a0 b0
a b 1 0
= .
c d c0 d0 0 1
0 0
Let ψ(z) = ac0 z+d
z+b
0 . Then φ(ψ(z)) = z, and by reversing the roles of φ and ψ, we
have ψ(φ(z)) = z. Therefore φ maps the complex upper half-plane bijectively
onto itself with inverse ψ.
2
,Exercise 1.2.5 Show that the intersection of the hyperboloid x2 − y 2 − z 2 = 1
with a Euclidean plane passing through the origin is either empty or a hyperbola.
Solution: The equation of a plane passing through the origin is
ax + by + cz = 0, with (a, b, c) 6= (0, 0, 0).
Assume first that a = 0. Then by+cz = 0. Now, the hyperboloid x2 −y 2 −z 2 = 1
is symmetric with respect to the x-axis. Hence, we can rotate the normal vector
(b, c) in the yz-plane so that b = 0. Then z = 0 is the equation of the plane
and the intersection with the hyperboloid is the hyperbola x2 − y 2 = 1 in the
xy-plane.
Now assume a 6= 0. Then we can normalize the normal vector (a, b, c) so
that a = 1. Then the plane has the equation x+by +cz = 0. Next, we rotate the
normal vector (1, b, c) about the x-axis so that c = 0. Then the plane has the
equation x + by = 0. Hence, the intersection of the plane with the hyperboloid
satisfies the equation b2 y 2 − y 2 − z 2 = 1. Now, the equation (b2 − 1)y 2 − z 2 = 1
has a real solution if and only if b2 > 1. The intersection of the plane with the
hyperboloid is the set
{(−by, y, z) : (b2 − 1)y 2 − z 2 = 1}.
Let u = ( √b−b
2 +1
, √b12 +1 , 0) and v = (0, 0, 1). Then {u, v} is an orthonormal basis
√
for the plane x + by = 0. Let w = b2 + 1y. Then the intersection is the set
2
{wu + zv : bb2 −1 2 2
+1 w − z = 1}.
Hence, the intersection is either empty or a hyperbola.
1.3 Euclidean n-Space
Exercise 1.3.1 Let v0 , . . . , vm be vectors in Rn such that v1 − v0 , . . . , vm − v0
are linearly independent. Show that there is a unique m-plane of E n containing
v0 , . . . , vm . Conclude that there is a unique 1-plane of E n containing any two
distinct points of E n .
Solution: The vectors v0 , . . . , vm are contained in the m-plane
P = v0 + Span{v1 − v0 , . . . , vm − v0 }.
Suppose v0 , . . . , vm are contained in the m-plane a+V with V an m-dimensional
vector subspace of Rn . Then vi − v0 is in V for each i = 1, . . . , m. Hence
V = Span{v1 − v0 , . . . , vm − v0 },
since v1 − v0 , . . . , vm − v0 are linearly independent. As v0 is in a + V , we have
that v0 = a + v for some v in V . Hence v0 − a is in V . Therefore a + V = v0 + V .
Thus, the m-plane P = v0 + V is unique.
3
,Exercise 1.3.2 A line of E n is defined to be a 1-plane of E n . Let x, y be
distinct points of E n . Show that the unique line of E n containing x and y is
the set
{x + t(y − x) : t ∈ R}.
The line segment in E n joining x to y is defined to be the set
{x + t(y − x) : 0 ≤ t ≤ 1}.
Conclude that every line segment in E n extends to a unique line of E n .
Solution: This follows from Exercise 1.3.1, since
{x + t(y − x) : t ∈ R} = x + Span{y − x}.
Exercise 1.3.3 Two m-planes of E n are said to be parallel if and only if they
are cosets of the same m-dimensional vector subspace of Rn . Let x be a point
of E n outside of an m-plane P of E n . Show that there is a unique m-plane of
E n containing x parallel to P .
Solution: Suppose P = a+V with V an m-dimensional vector subspace of Rn .
The cosets of V partition Rn . Hence, there is a unique coset b + V containing
x. Then Q = b + V is the unique m-plane of E n containing x parallel to P .
Exercise 1.3.4 Two m-planes of E n are said to be coplanar if and only if there
is an (m + 1)-plane of E n containing both m-planes. Show that two distinct
m-planes of E n are parallel if and only if they are coplanar and disjoint.
Solution: Suppose P and Q are distinct parallel m-planes of E n . Then
P = a + V and Q = b + V with V an m-dimensional vector subspace of Rn and
b − a not in V . Let W = Span{b − a, V }. Then dim W = dim V + 1. Observe
that P, Q ⊂ a + W . Hence P and Q are coplaner and disjoint, since they are
distinct cosets of V .
Conversely, suppose P and Q are coplaner and disjoint. Then P, Q ⊂ c + W
with W an (m + 1)-dimensional vector subspace of Rn , and suppose P = a + U
and Q = b + V , with U and V m-dimensional vector subspaces of Rn . By
replacing P and Q with P − c and Q − c, we may assume that c = 0. Now
a + U ⊂ W implies a is in W and so U ⊂ W . Likewise b is in W and V ⊂ W .
By replacing P and Q with P − a and Q − a, we may assume a = 0. Now U
and b + V are disjoint and so b is not in V . Then W = Span{b, V }. Suppose
there is a u in U that is not in V . Write u = tb + v with t in R and v in V .
Then t 6= 0. Hence ut = b + vt and so U meets b + V , which is a contradiction.
Therefore U ⊂ V , and so U = V , since dim U = dim V . Hence P and Q are
parallel.
4
, Exercise 1.3.5 The orthogonal complement of an m-dimensional vector sub-
space V of Rn is defined to be the set
V ⊥ = {x ∈ Rn : x · y = 0 for all y in V }.
Prove that V ⊥ is an (n − m)-dimensional vector subspace of Rn and that each
vector x in Rn can be written uniquely as x = y + z with y in V and z in V ⊥ .
In other words, Rn = V ⊕ V ⊥ .
Solution: Let v1 , . . . , vm be a basis of V . Extend v1 , . . . , vm to a basis
v1 , . . . , vm , . . . , vn of Rn . By the Gram-Schmidt process, we may assume that
v1 , . . . , vn is an orthonormal basis. Then we have
Span{vm+1 , . . . , vn } ⊂ V ⊥ .
Pn
Let z be in V ⊥ . Then there are coefficients c1 , . . . , cn such that z = i=1 ci vi .
As z · vi = 0 for each i = 1, . . . , m, we have that ci = 0 for each i = 1, . . . , m.
Hence z is in Span{vm+1 , . . . , vn }. Therefore, we have
V ⊥ = Span{vm+1 , . . . , vn }.
Thus dim V ⊥ = n − m. Pn
in Rn . Then there P
If x isP are coefficients c1 , . . . , cn such that x = i=1 ci vi .
m n
Let y = i=1 ci vi and z = i=m+1 ci vi . Then x = y + z with y in V and z
in V ⊥ . Suppose x = y 0 + z 0 with y 0 in V and z 0 in V ⊥ . As y 0 · vi = ci = y · vi
for each i = 1, . . . , m, we have that y 0 = y, and as z 0 · vi = ci = z · vi for each
i = m + 1, . . . , n, we have that z 0 = z. Thus y and z are unique.
Exercise 1.3.6 Let P be a subset of E n . Prove that P is a hyperplane of E n
if and only if there is a unit vector u in Rn , which is unique up to sign, and a
real number s such that
P = {x ∈ E n : u · x = s}.
Solution: Suppose P is a hyperplane of E n . Then there exists a ∈ E n and an
(n − 1)-dimensional vector subspace V of Rn such that P = a + V . Let u be a
unit vector which is orthogonal to V . Then
V = hui⊥ = {x ∈ E n : u · x = 0}.
Hence
P = a+V = {a + x ∈ E n : u · x = 0}
= {x ∈ E n : u · (x − a) = 0}
= {x ∈ E n : u · x = u · a}.
Conversely, suppose u is a unit vector in E n and s is a real number such
that
P = {x ∈ E n : u · x = s}.
5
Third Edition
Solution Manual
John G. Ratcliffe
October 3, 2022
,Contents
1 Euclidean Geometry 1
1.1 Euclid’s Parallel Postulate . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Independence of the Parallel Postulate . . . . . . . . . . . . . . . 1
1.3 Euclidean n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Spherical Geometry 22
2.1 Spherical n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2 Elliptic n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Spherical Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Spherical Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 Spherical Trigonometry . . . . . . . . . . . . . . . . . . . . . . . 36
3 Hyperbolic Geometry 42
3.1 Lorentzian n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Hyperbolic n-Space . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Hyperbolic Arc Length . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Hyperbolic Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.5 Hyperbolic Trigonometry . . . . . . . . . . . . . . . . . . . . . . 62
4 Inversive Geometry 71
4.1 Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2 Stereographic Projection . . . . . . . . . . . . . . . . . . . . . . . 74
4.3 Möbius Transformations . . . . . . . . . . . . . . . . . . . . . . . 76
4.4 Poincaré Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.5 The Conformal Ball Model . . . . . . . . . . . . . . . . . . . . . 85
4.6 The Upper Half-Space Model . . . . . . . . . . . . . . . . . . . . 89
4.7 Classification of Transformations . . . . . . . . . . . . . . . . . . 94
5 Isometries of Hyperbolic Space 107
5.1 Topological Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Groups of Isometries . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.3 Discrete Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
i
, 5.4 Discrete Euclidean Groups . . . . . . . . . . . . . . . . . . . . . . 126
5.5 Elementary Groups . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6 Geometry of Discrete Groups 137
6.1 The Projective Disk Model . . . . . . . . . . . . . . . . . . . . . 137
6.2 Convex Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.3 Convex Polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.4 Geometry of Convex Polyhedra . . . . . . . . . . . . . . . . . . . 147
6.5 Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.6 Fundamental Domains . . . . . . . . . . . . . . . . . . . . . . . . 155
6.7 Convex Fundamental Polyhedra . . . . . . . . . . . . . . . . . . . 157
6.8 Tessellations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
7 Classical Discrete Groups 166
7.1 Reflection Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
7.2 Simplex Reflection Groups . . . . . . . . . . . . . . . . . . . . . . 169
7.3 Generalized Simplex Reflection Groups . . . . . . . . . . . . . . . 175
7.4 The Volume of a Simplex . . . . . . . . . . . . . . . . . . . . . . 178
7.5 Crystallographic Groups . . . . . . . . . . . . . . . . . . . . . . . 180
7.6 Torsion-Free Linear Groups . . . . . . . . . . . . . . . . . . . . . 182
8 Geometric Manifolds 185
8.1 Geometric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
8.2 Clifford-Klein Space-Forms . . . . . . . . . . . . . . . . . . . . . 188
8.3 (X, G)-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.4 Developing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.5 Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9 Geometric Surfaces 202
9.1 Compact Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.2 Gluing Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
9.3 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . 204
9.4 Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.5 Closed Euclidean Surfaces . . . . . . . . . . . . . . . . . . . . . . 212
9.6 Closed Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
9.7 Closed Hyperbolic Surfaces . . . . . . . . . . . . . . . . . . . . . 220
9.8 Hyperbolic Surfaces of Finite Area . . . . . . . . . . . . . . . . . 224
10 Hyperbolic 3-Manifolds 231
10.1 Gluing 3-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 231
10.2 Complete Gluing of 3-Manifolds . . . . . . . . . . . . . . . . . . . 236
10.3 Finite Volume Hyperbolic 3-Manifolds . . . . . . . . . . . . . . . 237
10.4 Hyperbolic Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 242
10.5 Hyperbolic Dehn Surgery . . . . . . . . . . . . . . . . . . . . . . 248
ii
,11 Hyperbolic n-Manifolds 252
11.1 Gluing n-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 252
11.2 Poincaré’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 256
11.3 The Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . 258
11.4 Simplices of Maximum Volume . . . . . . . . . . . . . . . . . . . 260
11.5 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 265
11.6 Simplicial Volume . . . . . . . . . . . . . . . . . . . . . . . . . . 271
11.7 Measure Homology . . . . . . . . . . . . . . . . . . . . . . . . . . 274
11.8 Mostow Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
12 Geometrically Finite n-Manifolds 288
12.1 Limit Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
12.2 Limit Sets of Discrete Groups . . . . . . . . . . . . . . . . . . . . 289
12.3 Limit Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
12.4 Geometrically Finite Discrete Groups . . . . . . . . . . . . . . . 298
12.5 Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 303
12.6 The Margulis Lemma . . . . . . . . . . . . . . . . . . . . . . . . 304
12.7 Geometrically Finite Manifolds . . . . . . . . . . . . . . . . . . . 311
12.8 Arithmetic Hyperbolic Groups . . . . . . . . . . . . . . . . . . . 315
13 Geometric Orbifolds 320
13.1 Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
13.2 (X, G)-Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
13.3 Developing Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . 326
13.4 Gluing Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
13.5 Poincaré’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 336
iii
,Chapter 1
Euclidean Geometry
1.1 Euclid’s Parallel Postulate
1.2 Independence of the Parallel Postulate
Exercise 1.2.1 Let P be a point outside a line L in the projective disk model.
Show that there exists two lines L1 and L2 passing through P parallel to L such
that every line passing through P parallel to L lies between L1 and L2 . The two
lines L1 and L2 are called the parallels to L at P . All the other lines passing
through P parallel to L are called ultraparallels to L at P . Conclude that there
are infinitely many ultraparallels to L at P .
Solution: The lines L1 and L2 are the two lines passing through P that end
at the two ideal endpoints of L. See Figure 1.2.1. There are obviously infinitely
many lines in the projective disk model passing through P lying between L1
and L2 , and so there are infinitely many ultraparallels to L at P .
Exercise 1.2.2 Prove that any triangle in the conformal disk model, with a
vertex at the center of the model, has angle sum less than 180◦ .
Solution: Let ∆ be a triangle with vertices A, B, C in the conformal disk model
with C the center of the model. The sides AC and BC of ∆ are Euclidean line
segments. Let L be the hyperbolic line passing through the points A and B.
Then L is a circular arc that is orthogonal to the circle at infinity. The hyperbolic
half-plane bounded by L that does not contain ∆ is Euclidean convex, since it
is the intersection of two disks. Therefore, side AB of ∆ is a circular arc that
is contained in the corresponding Euclidean triangle ∆0 with vertices A, B, C.
The angle of ∆ at A is the angle between side AC and the Euclidean tangent
line T to the circular arc AB at A. The angle between line T and the Euclidean
line segment AB at A is positive. Hence, the angle of ∆ at A is less than the
angle of ∆0 at A. Likewise, the angle of ∆ at B is less than the angle of ∆0 at
1
,B. Therefore, the sum of the angles of ∆ is less than the sum of the angles of
∆0 , and so the sum of the angles of ∆ is less than 180◦ .
Exercise 1.2.3 Let u, v be distinct points of the upper half-plane model. Show
how to construct the hyperbolic line joining u and v with a Euclidean ruler and
compass.
Solution: If Re(u) = Re(v), draw the vertical ray L starting at Re(u) and
passing through u and v. Otherwise, draw the Euclidean line segment S joining
the points u and v. Construct the Euclidean perpendicular bisector M of the
line segment S. Let w be the point of intersection of M and the real axis. Then
|u − w| = |v − w|. Draw the semi-circle L centered at w passing through the
points u and v with endpoints on the real axis. Then in either case L is the
hyperbolic line joining u to v.
Exercise 1.2.4 Let φ(z) = az+b
cz+d with a, b, c, d in R and ad − bc > 0. Prove that
φ maps the complex upper half-plane bijectively onto itself.
Solution: Observe that
az + b cz + d ac|z|2 + adz + bcz + bd
· =
cz + d cz + d |cz + d|2
2
ac|z| + (ad + bc)Re(z) + (ad − bc)Im(z)i + bd
=
|cz + d|2
Hence, we have
az + b (ad − bc)Im(z)
Im = > 0.
cz + d |cz + d|2
Therefore φ maps the complex upper half-plane into itself.
Define real numbers a0 , b0 , c0 , d0 by the matrix equation
0 −1
a b0
a b
= .
c0 d0 c d
Observe that
a0 z+b0 aa0 z+ab0 bc0 z+bd0
a c0 z+d0 +b c0 z+d0 + c0 z+d0
=
a0 z+b0 ca0 z+cb0 dc0 z+dd0
c c0 z+d0 +d c0 z+d0 + c0 z+d0
(aa0 + bc0 )z + (ab0 + bd0 )
=
(ca0 + dc0 )z + (cb0 + dd0 )
= z,
since
a0 b0
a b 1 0
= .
c d c0 d0 0 1
0 0
Let ψ(z) = ac0 z+d
z+b
0 . Then φ(ψ(z)) = z, and by reversing the roles of φ and ψ, we
have ψ(φ(z)) = z. Therefore φ maps the complex upper half-plane bijectively
onto itself with inverse ψ.
2
,Exercise 1.2.5 Show that the intersection of the hyperboloid x2 − y 2 − z 2 = 1
with a Euclidean plane passing through the origin is either empty or a hyperbola.
Solution: The equation of a plane passing through the origin is
ax + by + cz = 0, with (a, b, c) 6= (0, 0, 0).
Assume first that a = 0. Then by+cz = 0. Now, the hyperboloid x2 −y 2 −z 2 = 1
is symmetric with respect to the x-axis. Hence, we can rotate the normal vector
(b, c) in the yz-plane so that b = 0. Then z = 0 is the equation of the plane
and the intersection with the hyperboloid is the hyperbola x2 − y 2 = 1 in the
xy-plane.
Now assume a 6= 0. Then we can normalize the normal vector (a, b, c) so
that a = 1. Then the plane has the equation x+by +cz = 0. Next, we rotate the
normal vector (1, b, c) about the x-axis so that c = 0. Then the plane has the
equation x + by = 0. Hence, the intersection of the plane with the hyperboloid
satisfies the equation b2 y 2 − y 2 − z 2 = 1. Now, the equation (b2 − 1)y 2 − z 2 = 1
has a real solution if and only if b2 > 1. The intersection of the plane with the
hyperboloid is the set
{(−by, y, z) : (b2 − 1)y 2 − z 2 = 1}.
Let u = ( √b−b
2 +1
, √b12 +1 , 0) and v = (0, 0, 1). Then {u, v} is an orthonormal basis
√
for the plane x + by = 0. Let w = b2 + 1y. Then the intersection is the set
2
{wu + zv : bb2 −1 2 2
+1 w − z = 1}.
Hence, the intersection is either empty or a hyperbola.
1.3 Euclidean n-Space
Exercise 1.3.1 Let v0 , . . . , vm be vectors in Rn such that v1 − v0 , . . . , vm − v0
are linearly independent. Show that there is a unique m-plane of E n containing
v0 , . . . , vm . Conclude that there is a unique 1-plane of E n containing any two
distinct points of E n .
Solution: The vectors v0 , . . . , vm are contained in the m-plane
P = v0 + Span{v1 − v0 , . . . , vm − v0 }.
Suppose v0 , . . . , vm are contained in the m-plane a+V with V an m-dimensional
vector subspace of Rn . Then vi − v0 is in V for each i = 1, . . . , m. Hence
V = Span{v1 − v0 , . . . , vm − v0 },
since v1 − v0 , . . . , vm − v0 are linearly independent. As v0 is in a + V , we have
that v0 = a + v for some v in V . Hence v0 − a is in V . Therefore a + V = v0 + V .
Thus, the m-plane P = v0 + V is unique.
3
,Exercise 1.3.2 A line of E n is defined to be a 1-plane of E n . Let x, y be
distinct points of E n . Show that the unique line of E n containing x and y is
the set
{x + t(y − x) : t ∈ R}.
The line segment in E n joining x to y is defined to be the set
{x + t(y − x) : 0 ≤ t ≤ 1}.
Conclude that every line segment in E n extends to a unique line of E n .
Solution: This follows from Exercise 1.3.1, since
{x + t(y − x) : t ∈ R} = x + Span{y − x}.
Exercise 1.3.3 Two m-planes of E n are said to be parallel if and only if they
are cosets of the same m-dimensional vector subspace of Rn . Let x be a point
of E n outside of an m-plane P of E n . Show that there is a unique m-plane of
E n containing x parallel to P .
Solution: Suppose P = a+V with V an m-dimensional vector subspace of Rn .
The cosets of V partition Rn . Hence, there is a unique coset b + V containing
x. Then Q = b + V is the unique m-plane of E n containing x parallel to P .
Exercise 1.3.4 Two m-planes of E n are said to be coplanar if and only if there
is an (m + 1)-plane of E n containing both m-planes. Show that two distinct
m-planes of E n are parallel if and only if they are coplanar and disjoint.
Solution: Suppose P and Q are distinct parallel m-planes of E n . Then
P = a + V and Q = b + V with V an m-dimensional vector subspace of Rn and
b − a not in V . Let W = Span{b − a, V }. Then dim W = dim V + 1. Observe
that P, Q ⊂ a + W . Hence P and Q are coplaner and disjoint, since they are
distinct cosets of V .
Conversely, suppose P and Q are coplaner and disjoint. Then P, Q ⊂ c + W
with W an (m + 1)-dimensional vector subspace of Rn , and suppose P = a + U
and Q = b + V , with U and V m-dimensional vector subspaces of Rn . By
replacing P and Q with P − c and Q − c, we may assume that c = 0. Now
a + U ⊂ W implies a is in W and so U ⊂ W . Likewise b is in W and V ⊂ W .
By replacing P and Q with P − a and Q − a, we may assume a = 0. Now U
and b + V are disjoint and so b is not in V . Then W = Span{b, V }. Suppose
there is a u in U that is not in V . Write u = tb + v with t in R and v in V .
Then t 6= 0. Hence ut = b + vt and so U meets b + V , which is a contradiction.
Therefore U ⊂ V , and so U = V , since dim U = dim V . Hence P and Q are
parallel.
4
, Exercise 1.3.5 The orthogonal complement of an m-dimensional vector sub-
space V of Rn is defined to be the set
V ⊥ = {x ∈ Rn : x · y = 0 for all y in V }.
Prove that V ⊥ is an (n − m)-dimensional vector subspace of Rn and that each
vector x in Rn can be written uniquely as x = y + z with y in V and z in V ⊥ .
In other words, Rn = V ⊕ V ⊥ .
Solution: Let v1 , . . . , vm be a basis of V . Extend v1 , . . . , vm to a basis
v1 , . . . , vm , . . . , vn of Rn . By the Gram-Schmidt process, we may assume that
v1 , . . . , vn is an orthonormal basis. Then we have
Span{vm+1 , . . . , vn } ⊂ V ⊥ .
Pn
Let z be in V ⊥ . Then there are coefficients c1 , . . . , cn such that z = i=1 ci vi .
As z · vi = 0 for each i = 1, . . . , m, we have that ci = 0 for each i = 1, . . . , m.
Hence z is in Span{vm+1 , . . . , vn }. Therefore, we have
V ⊥ = Span{vm+1 , . . . , vn }.
Thus dim V ⊥ = n − m. Pn
in Rn . Then there P
If x isP are coefficients c1 , . . . , cn such that x = i=1 ci vi .
m n
Let y = i=1 ci vi and z = i=m+1 ci vi . Then x = y + z with y in V and z
in V ⊥ . Suppose x = y 0 + z 0 with y 0 in V and z 0 in V ⊥ . As y 0 · vi = ci = y · vi
for each i = 1, . . . , m, we have that y 0 = y, and as z 0 · vi = ci = z · vi for each
i = m + 1, . . . , n, we have that z 0 = z. Thus y and z are unique.
Exercise 1.3.6 Let P be a subset of E n . Prove that P is a hyperplane of E n
if and only if there is a unit vector u in Rn , which is unique up to sign, and a
real number s such that
P = {x ∈ E n : u · x = s}.
Solution: Suppose P is a hyperplane of E n . Then there exists a ∈ E n and an
(n − 1)-dimensional vector subspace V of Rn such that P = a + V . Let u be a
unit vector which is orthogonal to V . Then
V = hui⊥ = {x ∈ E n : u · x = 0}.
Hence
P = a+V = {a + x ∈ E n : u · x = 0}
= {x ∈ E n : u · (x − a) = 0}
= {x ∈ E n : u · x = u · a}.
Conversely, suppose u is a unit vector in E n and s is a real number such
that
P = {x ∈ E n : u · x = s}.
5
