THOMAS KOBERDA

  1.   Free groups and free products                                        1
  2.   Basics of hyperbolic geometry and background from differential
       geometry                                                             3
  3.   Quasi-isometries                                                     8
  4.   Geometrization of surfaces (INCOMPLETE)                             11
  5.   The geometry of H2 /P SL2 (Z)                                       11
  6.   HNN extensions and embedding theorems for finitely generated
       groups                                                              13
  7.   Unsolvability of certain decision problems in group theory          15

  These course notes will follow the material covered in lecture. They will
be updated regularly as the course progresses.
  The reader is assumed to have a basic knowledge of group theory, topology
and geometry, equivalent to that of an intermediate-to-advanced undergrad-
uate. The material here is supplemented by the extensive problems.

                  1. Free groups and free products
   Let G be a group. If we forget the operation, we obtain a set. This is
called the “forgetful functor” in the language of category theory. We wish
to construct an left adjoint to this functor, called the “free group functor”.
In less nonsensical terminology, we are given a group G and a set X, and
we want a group F (X) such that for each map of sets
                                 f :X→G
we obtain a unique group homomorphism
                             F (f ) : F (X) → G
which extends f . Intuition tells us that F (X) should just be the set of
words in X and the formal inverses of elements of X under concatenation,
and that F (f ) should be defined in the unique way that makes this map a
  Let w be a formal word in X and its formal inverses. We say that w is
reduced if there is no occurrence of xx−1 or x−1 x as a subword of w.
2                               T. KOBERDA

Theorem 1.1 (Free groups exist). The free group on a set X, denoted
F (X) consists of the set of reduced words in X and its formal inverses.
The identity (1) is the empty word. The group operation is given by formal
   The reason that this is a theorem is that we need to do some work to
prove that our intuitive definition really does give rise to a group law. The
existence of the identity is be definition and its properties are immediate.
The existence of inverses is also obvious, since we take a word in X, write
it backwards, and replace each letter with its inverse. The difficulty is in
verifying that the concatenation operation is associative.
   If we can embed the set of reduced words in X into a group of automor-
phisms of some object, then the associativity of the composition of permu-
tations does all the work for us.
Proof that free groups exist. We will make F (X) act on a certain graph. Let
the vertices of a graph T (X) be the reduced words in X. We draw a single
edge between two vertices if the two words differ by multiplication by an
element of X on the right. We label the edge by the element by which we
multiply. Note that this is not quite well defined: to traverse an edge in one
direction we multiply by an element x ∈ X and in the other direction we
multiply by x−1 . The reader should verify that T (X) is a tree. To do this,
use the notion of length of words: for w ∈ F (X), (w) is the smallest n such
that w is a reduced product of n elements of X.
   For w ∈ F (X), we define a symmetry of T (X) which we denote by fw .
The action of fw is to multiply on the left by w and reduce. The reader
should verify that w acts without a fixed point if w = 1 and preserves the
edge relation and hence is an automorphism of T (X), and that thus we
embed F (X) as a group of automorphisms of T (X).
   In group theory, we are interested in three fundamental problems:
    (1) Given G = S | R and w ∈ F (S), can we algorithmically determine
         if w = 1 in G (word problem)?
    (2) Given g, h ∈ G, can we algorithmically determine if g and h are
         conjugate (conjugacy problem)? In the same automorphism orbit
         (Aut(G)-orbit problem)?
    (3) Given two groups G, H together with finite presentations, can we
         algorithmically determine whether G and H are isomorphic (iso-
         morphism problem)?
   For the free group we can easily answer all of these questions (we will
return to the automorphism orbit problem later). A word in a free group (it
must be presented as a group with generators and no relations) is nontrivial
if and only if it is reduced and nontrivial. Two words are conjugate if and
only if they are cyclically reduced and equal. Two free groups are isomorphic
if and only if they have bases of the same cardinality, equivalently if their
abelianizations are isomorphic.
   In general, these problems are not solvable, even for finitely presented
   Now that free groups are on sound footing, we may speak of presentations
for groups. A presentation is an expression of the form G = S | R . The set
                                 COURSE NOTES                                    3

S is called the set of generators and R ⊂ F (S) is the set of relations. These
specify a group G by taking F (S) modulo the smallest normal subgroup
containing R. G is finitely generated if S is finite. It is finitely related if R
is finite. It is finitely presented if both S and R are finite.
   Given two groups A and B we want to make sense of their free product.
The universal property should be that if G is any group and φ, ψ are any two
homomorphisms A, B → G respectively, then there should exist a unique
A ∗ B and φ ∗ ψ extending φ and ψ. This group is called the free product.
Its existence is witnessed by taking A = SA | RA and B = SB | RB and
A ∗ B = SA , SB | RA , RB . Uniqueness is, as usual, a consequence of the
universal property.
   Every element w ∈ A ∗ B can be written uniquely (verify this!) as a
product of the form w = a1 b1 · · · an bn , where ai ∈ A, bi ∈ B, and each ai , bi
(except for possibly a1 and bn ) is nontrivial.
   There is a different useful characterization of free products:
Lemma 1.2 (Ping-pong lemma). Let G act on a set X and let A, B < G be
two subgroups which together generate G. Let X1 , X2 be nonempty subsets
of X which are disjoint. Suppose that g(X2 ) ⊂ X1 for all g ∈ A and
g(X1 ) ⊂ X2 for all g ∈ B, and suppose that |A| ≥ 3, |B| ≥ 2. Then
G ∼ A ∗ B.
Proof. Assume w = a1 b1 · · · ak ∈ A ∗ B, each letter nontrivial in A or B.
Observing w(X2 ) ⊂ X1 shows w is nontrivial. If w = b1 a2 · · · bk , conjugate
by an element of A and apply the previous observation. If w = a1 b1 · · · bk ,
choose a ∈ A with a = a−1 to use the previous observation. The remaining
case is analogous.
Corollary 1.3. The matrices
                                 1 2   1 0
                                 0 1   2 1
generate a free group.
Proof. Use the usual action of SL2 (Z) on R2 , and use as X1 and X2 the
regions above and below the line y = x.

     2. Basics of hyperbolic geometry and background from
                       differential geometry
  We proved in class that the stabilizer of H2 in Aut(C) is P SL2 (R).
Theorem 2.1 (Classification of elements of P SL2 (R)). Every element A ∈
P SL2 (R) is conjugate to a diagonal matrix, a unipotent matrix which is
upper triangular, or a rotation. This can be determined by looking at the
absolute value of the trace of A and determining whether it is larger, equal
to, or less than 2, respectively.
Proof. Since we are consider the absolute value of the trace, we will have no
problems lifting A to SL2 (R). The characteristic polynomial pA is given by
x2 − tr(A)x + 1. The discriminant of pA is (tr(A)2 − 4). In the first case, this
number is positive so that A is diagonalizable, which was the first claim. In
the second case, the discriminant if 0 so that pA has a unique root which
4                                  T. KOBERDA

is 1 (up to a sign) so that A is conjugate to a unipotent upper triangular
matrix by the Jordan normal form. In the last case, pA has two conjugate
non-real roots. The reader should verify that then A has a fixed point in
H2 , which up to conjugacy is i (or the origin in the disk model of hyperbolic
space). The Schwarz lemma gives our conclusion.
   We want to describe simply connected Riemann surfaces together with
intrinsic metrics which are invariant under their groups of holomorphic au-
tomorphisms. The Riemann Mapping Theorem says that any simply con-
nected Riemann surface is biholomorphic to either the Riemann sphere C,
the complex plane C, or upper half space H2 . The fact that C is not con-
tractible implies that it is not biholomorphic to either of the other two.
Liouville’s Theorem implies that any holomorphic map C → H2 is constant.
   Let α be a holomorphic automorphism of C. Let r1 , . . . , rn be its roots
and p1 , . . . , pm its poles (it is a nontrivial fact that n = m). Comparing α
                                             (z − ri )
                                   q(z) =
                                             (z − pj )
implies by Liouville’s Theorem that α and q differ by a nonzero constant
multiple. It is only possible for α to by a bijection if both the numerator
and denominator have degree one, and one verifies that each α ∈ Aut(C) is
a fractional linear transformation.
   If α ∈ Aut(C), then the removable singularity theorem imlies that α
extends to an automorphism of C which fixes ∞ and is therefore an affine
transformation. If α ∈ Aut(H2 ), we use the fact that H2 is biholomorphic
to the unit disk, so that the Schwarz reflection principle shows that we
can holomorphically extend α to C by 1/α(z). It follows that P SL2 (R) is
precisely the holomorphic automorphism group of H2 .
   Let M be a Riemannian manifold. This means that we have a smoothly
varying family of positive-definite inner products gx : Tx M × Tx M → R.
Abstractly, these inner products are suppose to give a notion of length to
each tangent vector at x ∈ M . The positive-definite condition says that
for each x and v ∈ Tx M , gx (v, v) ≥ 0 with equality holding if and only if
v = 0. If f : M → M is smooth map, we say that f preserves g if g = f ∗ g.
Precisely what this means is that gx (v, v) = gf (x) (Df (v), Df (v)).
   In classical differential geometry, if U ⊂ R2 is an open set with coordinates
x, y, then a metric is specified by a 2-tensor E(x, y) dx2 + 2F (x, y) dxdy +
G(x, y) dy 2 (this notation goes back to Gauss). In more modern terminology,
this expression should be thought of as a smooth section of Sym2 (T ∗ M ),
the second symmetric power of the cotangent bundle, but this level of ab-
straction is unnecessary for our purposes.
   Fixing a point (x, y) ∈ R2 , we can reconstruct the Riemannian metric
from the 2-tensor above by putting the terms into a matrix:
                                    E(x, y) F (x, y)
                       g(x, y) =                     .
                                    F (x, y) G(x, y)
The inner product of two vectors (a, b) and (c, d) at (x, y) is given by ap-
plying the linear map g(x, y) to the matrix (c, d) and then taking the usual
inner product of the resulting vector with (a, b). One should verify that the
                                 COURSE NOTES                                 5

resulting real number is unchanged by switching the role of (a, b) and (c, d),
and this follows from the easy observation that A is self-adjoint.
   Let g1 and g2 be Riemannian metrics on M . We say that g1 and g2 are
conformal, or are in the same conformality class, if there is an everywhere
positive function M → R which satisfies g1 = f (x)g2 . It is obvious that
there can exist metrics which are not in the same conformality class when
the dimension of M ≥ 2, since it is not true that all positive-definite inner
products are multiples of each other. We call the metrics on U which are in
the conformality class of the Euclidean metric conformal.
   If we identify U ⊂ C, the metric becomes expressible as the symmetric 2-
tensor ϕ(z) dz 2 = E(x, y)(dx2 +dy 2 ). The reader should verify that there are
no conformal metrics on C or C which are invariant under all holomorphic
automorphisms of the respective Riemann surfaces.
Theorem 2.2. The metric
                                   1 − |z|2
is the unique (up to scale) conformal metric on the unit disk in C which is
invariant under all of its holomorphic automorphisms. This metric is called
the hyperbolic metric.
Proof. Let ϕ(0) > 0 be any candidate. There is a holomorphic automor-
phism of the disk which sends a particular z to zero, and one such is given
                                       ζ −z
                                 ζ→           .
                                       1 − zζ
If v is a tangent vector at z, we see that v must be sent to
                                   1 − |z|2
which can be seen by computing the derivative. Since the metric at z is the
pullback of the metric at zero, we must have
                               ϕ(z) =            .
                                        1 − |z|2

Corollary 2.3. The hyperbolic metric on H2 is given by
or in Euclidean coordinates by
Proof. We use the fact that H2 is biholomorphic to the unit disk via the
                               z→         ,
and hence the hyperbolic metric on H2 must be the pullback of the holo-
morphic metric on the unit disk via this biholomorphism.
6                                 T. KOBERDA

  We want to develop the appropriate notion of lines in H2 , which are called
geodesics. They are locally distance minimizing.
Theorem 2.4. Let b > a > 0. The hyperbolic distance between bi and ai is
Proof. Let ai and bi be connected by a path (x(t), y(t)) parametrized over
the unit interval. We have a sequence of inequalities:
                           b            1
                              dy             (dx/dt)2 + (dy/dt)2
              ln(b/a) =           ≤                              dt.
                            a y       0              y
The last expression is the hyperbolic length of (x(t), y(t)), and the second
to last is the result if x(t) = 0 for all t.
   The reader should fill in the details necessary to establish that the vertical
segment connecting ai and bi is the unique such shortest path.
   Therefore, straight vertical lines in H2 are geodesics. Since P SL2 (R) acts
by isometries, we see that semicircles which intersect the real line perpen-
dicularly are all geodesics in the hyperbolic metric. If x, y ∈ H2 are any two
points, then they lie on either a straight vertical line or on such a semicircle.
Therefore any two points in H2 can be connected by a unique geodesic. It
is easy at this point to see that a hyperbolic element of P SL2 (R) stabilizes
a unique geodesic in H2 .
   We are now in a position to give another interesting application of the
ping-pong lemma.
Theorem 2.5. Let A and B be two hyperbolic elements in P SL2 (R) which
fix different geodesics in H2 . Then there exists an N > 0 such that AN , B N
is free.
Proof. Consider the four endpoints of the two geodesics. Cut out four dis-
joint half-planes about each of the endpoints. By considering the dynamics
of a hyperbolic element we see that one of the endpoints of each geodesic
is a repelling fixed point and the other is an attracting fixed point for pos-
itive powers of the stabilizing hyperbolic elements. We see that sufficiently
high positive powers of A send everything except for the half-plane contain-
ing the repelling fixed point of A into the half-plane containing containing
the attracting fixed point. It is easy to see that the ping-pong lemma now
   As an application of hyperbolic geometry to group theory, we prove the
following theorem which is a reflection of more general properties of so-called
hyperbolic groups:
Theorem 2.6. A free group F has the unique roots property: if g, h ∈ F
and n ∈ Z, we consider the equation g n = h. If a solution exists, it is
Proof. We do the case where F has rank two and leave the general case to
the reader. We can embed F as a subgroup of P SL2 (R). If h ∈ F , then as
an element of P SL2 (R) it is either elliptic, parabolic or hyperbolic. Suppose
that g exists. It is easy to check that p ∈ H2 ∪ R ∪ {∞} is a fixed point
of h if and only if it is a fixed point of g. It follows that g has the same
                               COURSE NOTES                                   7

classification as h. An elliptic element is classified by its fixed point in H2
and the degree by which it rotates (after we conjugate the fixed point to the
origin in the unit disk). A parabolic element is classified by its fixed point at
infinity and its translation distance along the x-axis in the upper half-plane
model (after we conjugate the fixed point to be at infinity). A hyperbolic
element is classified by its fixed points at infinity and its translation distance
along its fixed geodesic. In all but the elliptic case, the claim is clear. On
the other hand, we can always choose the embedding of F into P SL2 (R) to
avoid elliptic elements.
   Next we would like to calculate the areas of some naturally occurring
shapes in H2 . To define what these objects might be, we define a half-space
in H2 to be determined by a geodesic γ in H2 . The complement of γ in
H2 is a union of two regions. We define a convex polygon in H2 to be a
finite intersection of half-spaces. A polygon in H2 is a finite union of convex
polygons. The intersection of two geodesics bounding P is called a vertex of
P . The angle at the vertex is the Euclidean angle between the two tangent
vectors. If two geodesics bounding sides of P meet at a point at the circle at
infinity, then the point at infinity where they meet is called an ideal vertex
of P . The reader should verify that the angle at an ideal vertex is zero.
   If P is a polygon in H2 , then removing larger and larger balls centered at
a point in the interior of H2 yields a finite number of ends of the polygon.
Intuitively, they can be defined for any topological space X and are the
connected components of the complements of larger and larger compact sets
K ⊂ X. Rigorously, the ends of X, denoted e(X), are defined to by
                               lim π0 (X \ K).
                               ← −

If an end is a piece of the polygon bounded by two geodesics going off to
the same point at infinity, we say that the end is a cusp of P .
Proposition 2.7. The area of a polygon P is finite if and only if one of the
two following conditions hold:
   (1) P is bounded.
   (2) P is unbounded but all of its ends are cusps.
Proof. In the first case, P is compact and hence has finite area. For the
second case, we will find an upper bound for the area of a cusp in the
sequel. If P is unbounded and has an end which is not a cusp, then two
consecutive geodesics bounding sides of P (in the cyclic order about the
circle at infinity) do not meet at the same point at infinity. It is easy to
check that the resulting end must have infinite area.
   A triangle in P is an intersection of 3 half-spaces. The following theorem
is an amazing fact about hyperbolic space which is a reflection of its nonzero
curvature. Note that is definitely false in Euclidean geometry.
Theorem 2.8 (Gauss–Bonnet Theorem for hyperbolic triangles). Let T be
a hyperbolic triangle with one ideal vertex and two vertices in the interior of
H2 with angles α and β. Then the area of T is π − (α + β). In particular,
the area of an ideal triangle is π.
8                                        T. KOBERDA

Proof. We consider the upper half-plane model. Since P SL2 (R) acts by
isometries, we may assume the ideal vertex is at infinity and that the other
two vertices sit on a unit radius circle centered at the origin over the points
a and b on the x-axis. We have:
                                                         b   ∞
                                      1                                 1
                Area =                   dx dy =             √             dy dx.
                                      y2             a           1−x2   y2
Simplifying, we get
                              b                          β
                                  √          dx =            −1 dθ,
                          a           1 − x2         π−α
where we have made the substitution x = cos θ in the last step. The claim
    It is easy to deduce the following corollary, which we leave as an exercise.
Corollary 2.9. A hyperbolic triangle with angles α, β, γ has area π − (α +
β + γ).

                                  3. Quasi-isometries
   In this section we develop one of the most fundamental notions in geo-
metric group theory, namely that of quasi-isometry. One of the motivating
problems behind quasi-isometries is the fact that a presentation for a group
is not well-defined. To illustrate the problems this causes, we let G = S | R
be a finitely generated group. We define the word length
by taking all strings in S which represent a fixed word g. We define (g) to
be the shortest such string (it may not be unique, but there exists a string
of shortest length representing g). Consider two presentations for F2 , the
free group of rank two. One is given by
                                            x, y ,
and the other is given by
                                       a, b, c | c = an .
The reader should verify that both of these are presentations for F2 , and an
isomorphism is given by sending x → a and y → b. Notice that with respect
to the first presentation, the word xn has length n. Identifying xn with an
via the isomorphism, it has length 1 with respect to the second presentation.
   Let X and Y be metric spaces. We say that a function f : X → Y is a
quasi-isometry if there exist constants λ ≥ 1 and C ≥ 0 such that
                dX (x, y) − C ≤ dY (f (x), f (y)) ≤ λdX (x, y) + C.
We say that X and Y are quasi-isometric (via f ) if there is a D ≥ 0 such
that every point of Y is within a D-neighborhood of a point of f (X).
   Quasi-isometry is a rigorous expression of the notion that two spaces “look
the same from far away.” Therefore, the constant C is introduced to allow
for tearing on a small scale. Quasi-isometries are generally not continuous.
                               COURSE NOTES                                 9

   The reader should verify that any finite diameter metric space is quasi-
isometric to a point.
   We will be dealing with quasi-isometry often when either X or Y (or
both) are the Cayley graph of a finitely generated group. By the geometry
of a group we mean the geometry of the quasi-isometry type of its Cayley
graph. Recall that if G = S | R is a group, we construct a graph Γ(G, S)
whose vertices are the elements of G and two vertices are connected if they
differ by an element of S. Precisely, we have that g and h are connected is
there is an s ∈ S such that g = s · h. We will generally assume that S is
closed under taking inverses and does not contain the identity element. We
will partition S into two subsets S and (S )−1 , and we will label the edges
of Γ(G, S) by elements of S . There will be a positive orientation on the
edge from h to g if g = s · h. Traversing the edge in the other direction is
tantamount to multiplying on the left by s−1 .
   Note that loops in Γ(G, S) correspond precisely to strings in S which
represent the identity in G. Note also that the word length (g) coincides
with the graph distance dΓ (1, g) in the Cayley graph. As an exercise, the
reader should draw part of the Cayley graph for F2 with respect to the two
presentations given above.

Theorem 3.1. Let G admit presentations S1 | R1 and S2 | R2 with both
S1 and S2 finite. Then Γ(G, S1 ) and Γ(G, S2 ) are quasi-isometric.

Proof. Exercise.

   We now wish to state what is often called the fundamental observation
of geometric group theory, or the so-called Milnor-Svarc Lemma. Before
stating it, we need a few definitions. Let X be a metric space. We say that
X is proper if closed balls of finite radius are compact. We say that a path
γ between two points x, y in a metric space X satisfying d(x, y) = L is a
geodesic if γ is an isometric to [0, L]. A metric space is called geodesic if
every pair of points can be joined by at least one geodesic segment. A group
action G × X → X is said to be proper if for each compact K ⊂ X, the set

                          {g ∈ G | g · K ∩ K = ∅}

is finite.
Theorem 3.2 (Milnor-Svarc). Let X be a metric space which is geodesic
and proper, and let G × X → X be a proper group action by isometries.
Suppose furthermore that X/G is compact. Then G is finitely generated,
and for each x ∈ X, the map g → g · x is a quasi-isometry G → X.

Proof. The proof follows de la Harpe, who follows Milnor. Let π : X → X/G
be the canonical projection. There is a metric on X/G given by looking at
the distance between preimages of two points x and y. One needs properness
to show that this is a metric. Let R be the diameter of X/G. If x ∈ X is
fixed, let B be a closed R-ball about x. The totality of the translates of B
is a covering of X. Let

                    S = {s ∈ G | s = 1, s · B ∩ B = ∅}.
10                               T. KOBERDA

Note that S is closed under taking inverses and is finite since the action of
G is proper. Let
                      r = inf{d(B, g · B) | g ∈ G \ S},
and let
                            λ = max d(x, s · x).
   Firstly, observe that r > 0. Indeed, for any one g, the distance is positive,
so let r be a distance that works for a particular g. Let
                     T = {g ∈ G \ S | d(B, g · B) ≤ r }.
Then T is finite by properness and nonempty by definition, whence the
   Next, we claim that S generates G. Let g ∈ G \ S. Let k be defined by
                     R + (k − 1)r ≤ d(x, g · x) < R + kr.
The reader should check that k ≥ 1. Since X is geodesic, we can connect
x and g · x by a piece-wise geodesic path with endpoints {x, x1 , . . . , xn }
such that d(x, x1 ) < R and d(xi , xi+1 ) < r. Since translates of B cover X,
there is a gi ∈ G such that gi · B contains xi . If si = gi−1 gi , we can write
g = s1 · · · sn . The reader should verify that each si ∈ S.
  Finally, for all g ∈ G, we have
                     1                          1
                       d(x, g · x) ≤ dS (1, g) ≤ d(x, g · x) + 1.
                     λ                          r
Here, dS is the word length with respect to S \ {1}. By what we did to prove
that S generates G, we obtain
                                         1                 R
                        dS (1, g) ≤ k ≤ d(x, g · x) + 1 − .
                                         r                  r
The other inequality is clear. It follows easily that the map G → X given
by g → g · x is a quasi-isometry.
  One of our goals is exhibit some non-obvious quasi-isometries. One such
quasi-isometry will be that the fundamental group of a closed surface of
genus g > 1 is quasi-isometric to H2 . Before we do this, we should know
what the closed surfaces are.
Theorem 3.3. Let Σ be a closed orientable surface. The Σ is homeomorphic
to a surface of genus g ≥ 0. We call the surface of genus zero the sphere
and the surface of genus one the torus.
Proof. Suppose first that if γ ⊂ Σ is a simple closed loop then Σ \ γ is
disconnected. We claim that then Σ is homeomorphic to the sphere. Let
∆ be a finite cell decomposition of Σ and let T be a maximal tree in the
1-skeleton of ∆. Note that Σ \ T is connected. Let ∆∗ be the dual cell
decomposition of Σ and let Γ be a maximal subgraph of the 1-skeleton of
∆∗ subject to the condition Γ∩T = ∅. Then Γ is a tree. Indeed, observe that
Γ is connected. If Γ contains a loop then Γ separates Σ, which contradicts
the maximality of T . It follows that Σ is a union of small neighborhoods of
T and Γ and is hence homeomorphic to the sphere.
   Let Σ contain a simple loop γ which does not separate Σ which sits within
the 1-skeleton of Σ. Cutting along such a loop and gluing in two disks gives
                               COURSE NOTES                                 11

another surface Σ . Notice that the Euler characteristic of Σ is greater than
that of Σ by two. If Σ still has nonseparating loops, repeat this process,
which we call “cutting and patching”.
   Finally, we must show that this is a finite process. Note that Σ has a cell
decomposition coming from Σ, except that Σ has two more faces and one
extra vertex and edge for each vertex and edge appearing in γ. Since γ is
a circle, the number of extra vertices and edges is equal. Suppose that the
cell decomposition of Σ is not redundant, in the sense that each vertex has
degree at least three. Then the ratio of edges to vertices must then be at
least 3/2.
   Note that γ has two preimages in Σ . For each vertex appearing in γ, one
of the two vertices in the two preimages of γ must be non-redundant. By
deleting a redundant vertex, we decrease the number of edges and vertices by
one. If one preimage of γ has at most one non-redundant vertex, then we can
delete the entire preimage of γ collapse the “loose ends” of the 1-skeleton.
This process will decrease the total number of edges and vertices in the 1-
skeleton in Σ from the numbers in Σ. We call this process “simplification”.
   Therefore, we take Σ and perform the two processes above. Either the
number of edges goes off to infinity or remains bounded. If it remains
bounded then since some of the contributions to the edges in Σ are one of
the preimages of γ and since γ separates Σ , after finitely many cuts and
patches and simplifications we must obtain the sphere. If the number of
edges goes off to infinity, then the ratio of edges to vertices must tend to
one, a contradiction.

         4. Geometrization of surfaces (INCOMPLETE)
  The first result allows us to simplify the geometrization process for all
surfaces to geometrization for one single surface.

Theorem 4.1. Let G = π1 (Σg ) and let G < G have index n. Then G =
π1 (Σn(g−1)+1 ). In particular, a closed surface of arbitrary genus (greater
than two) covers a surface of genus 2. Analogously, let F be a free group of
rank d and let F < F have index n. Then F has rank n(d − 1) + 1. In
particular every finite rank free group is contained in a free group of rank 2,
and it can be embedded with finite index when it is nonabelian.

Proof. The Euler characteristic of Σg is 2 − 2g, and Euler characteristic is
multiplicative under taking covers. It follows that G is the fundamental
group of a surface whose Euler characteristic is n(2 − 2g). Writing the genus
as g , we see that g − 1 = n(g − 1), whence the claim. The assertion for
free groups is analogous. The claims about embeddings follow from the fact
that a surface or free group has a normal subgroup of every index.

                   5. The geometry of H2 /P SL2 (Z)
   Let X be a metric space and G a group acting properly and discontin-
uously on X. Let F be a closed domain (the closure of a nonempty open
set). We call F a fundamental domain for the action of G if
12                                 T. KOBERDA

                                        g · F = X.
   (2) F 0 ∩ g · F 0 = ∅ for all nonidentity elements of G.
  The boundary ∂F of F is F \ F 0 . The collection of translates of F by
G is called a tesselation of X. As an exercise, the reader should check
that the semi-annulus bounded by the circles |z| = 1 and |z| = 2 in the
hyperbolic plane is a fundamental domain for the isometry z → 2z. Note
that fundamental domains are not well-defined.
  We will produce a fundamental domain for the action of P SL2 (Z) on H2 .
Proposition 5.1. The matrices
                                  1 1   1 0
                                  0 1   1 1
generate P SL2 (Z).
Proof. It suffices to show that these matrices generate SL2 (Z). Recall that
SL2 (Z) acts naturally on Z2 . If
                                    a b
                             A=             ∈ SL2 (Z),
                                    c d
then the greatest common divisor of the entries in any column or row is one.
Running the Euclidean argument on any of these pairs of integers yields
(1, 0) or (0, 1). The reader should verify that it follows that a there are
matrices P and Q which are products of
                                  1 1   1 0
                                  0 1   1 1
and their inverses such that
                                            1 0
                                 P AQ =
                                            n 1
for some n.
  It follows that the isometries z → z + 1 and z → z+1 generate P SL2 (Z).
As an exercise, the reader should find an expression for z → −1 in terms of
these isometries. A fundamental domain for P SL2 (Z) is bounded between
−1/2 ≤ (z) ≤ 1/2 and above the circle |z| = 1.
Proposition 5.2. The closed domain above is a fundamental domain for
the action of P SL2 (Z).
Proof. Let 1 = A be as above. We have:
     |cz + d|2 = c2 |z|2 + 2 (z)cd + d2 > c2 + d2 − |cd| = (|c| − |d|)2 + |cd|.
The lower bound is at least one and is an integer. It follows that (A(z)) <
 (z). Replacing z and A by A(z) and A−1 gives a contradiction.
   Note that P SL2 (Z) acts on H2 with some fixed points. Recall that any
element of P SL2 (R) with a fixed point is elliptic, and any elliptic element of
P SL2 (Z) has finite order (why?) The reader should verify that the order of
any torsion element of P SL2 (Z) has order at most 3. The reader should also
                                   COURSE NOTES                                          13

verify that z → z−1 has order 3 and preserves the cube roots of unity on the
unit circle. The images of the cube roots of unity and i in H2 /P SL2 (Z) are
precisely the images of points of H2 which are fixed by finite order elements
of P SL2 (Z) (the so-called “orbifold points”). The reader should check that
the volume of H2 /P SL2 (Z) is π/3.
Proposition 5.3. The torsion elements of P SL2 (Z) generate it.
Proof. We have
                           1 −1         0 −1                1 −1
                           1 0          1 0                 0 1
                           0 −1          1 −1                   1 0
                                                      =             .
                           1 0           1 0                    1 1

  Note that if Γ < P SL2 (Z) is torsion free then its index is at least 6. Let
Γ(n) be the kernel of the map P SL2 (Z) → P SL2 (Z/nZ).
  The reader should verify that P SL2 (Z/2Z) ∼ S3 . As a more difficult
exercise, the reader should check that Γ(2) is free of rank 2.

      6. HNN extensions and embedding theorems for finitely
                         generated groups
  In this section we will discuss some basics concerning Higman-Neumann-
Neumann extensions (HNN extensions) and their applicability to embed-
dings of countable groups in certain finitely generated groups.
  Let G be a group and A, B < G subgroups together with an isomorphism
φ : A → B. The HNN extension of G given by φ is a group with the
                     Gφ = G, t | t−1 at = φ(a), a ∈ A .
We call G the base of the HNN extension. Let                      i   = ±1. We say that the
                           g0 , t 1 , . . . , t n , gn ,
n ≥ 0, is reduced if there is no sequence t−1 gi t with gi ∈ A or tgi t−1 with
gi ∈ B.
Lemma 6.1 (Britton’s Lemma). If n ≥ 1 and the sequence
                                  g0 , t 1 , . . . , t n , gn
is reduced, then
                                 g0 t 1 · · · t n gn = 1
in Gφ .
  An easy consequence of Britton’s Lemma is the following:
Lemma 6.2. Let Gφ be an HNN extension with base G. If g ∈ Gφ has finite
order then g is conjugate to an element of finite order in G.
14                                   T. KOBERDA

   A more difficult generalization of Britton’s Lemma gives a way of dis-
tinguishing two elements in an HNN extension based on a so-called normal
form. Choose coset representatives for A and B in G, with the identity
representing A and B. A sequence
                                 g0 , t 1 , . . . , t n , gn ,
is called a normal form if:
      (1) g0 ∈ G is arbitrary.
      (2) i = −1 implies gi is a representative of a coset of A in G.
      (3) i = 1 implies gi is a representative of a coset of B in G.
      (4) no subsequence of the form t , 1, t− occurs.
Lemma 6.3 (Normal Form Theorem for HNN extensions). The base group
G embeds in Gφ and any element of Gφ has a unique representative in
normal form.
  The easy construction of HNN extensions yields some amazing conse-
Theorem 6.4. Every countable group C can be embedded in a quotient G
of F2 . We may assume that the two generators of G have infinite order. G
will have n-torsion if and only if C does.
Proof. Let
                               C = c1 , . . . | s1 , . . . .
Write F = C ∗ a, b . The set {a, b−1 ab, b−2 ab2 , . . .} generates a free subgroup
of a, b (why?), as does {b, c1 a−1 ba, . . . , cn a−n ban , . . .}. Write
                  G = F, t | t−1 at = b, t−1 b−i abi t = ci a−i bai .
Then C embeds in G by the Normal Form Theorem, and the claim about
torsion follows from Britton’s Lemma.
Corollary 6.5. There exist uncountably many non-isomorphic quotients of
F2 .
Proof. For every nonempty set S of primes, take the direct sum of cyclic
groups of order p for p ∈ S. Applying the previous theorem gives the
Corollary 6.6. There exists a finitely generated group which contains each
finitely presented group as a subgroup.
Proof. The direct sum of all finitely presented groups is a countable group.

     Even more bizarre embedding theorems are true. For instance:
Theorem 6.7. Every countable group can be embedded in a six-generator
simple group.
  Certain HNN extensions have very strange properties. A group G is called
Hopfian if any epimorphism G → G is an isomorphism.
                               COURSE NOTES                                  15

Proposition 6.8. The (2, 3)–Baumslag-Solitar group
                           G = b, t | t−1 b2 t = b3
is not Hopfian.
Proof. Define φ : G → G by φ(t) = t and φ(b) = b2 . It is easy to check
that φ is well-defined and surjective. Furthermore, [t−1 bt, b] ∈ ker(φ) and
is nontrivial by Britton’s Lemma.
  A group is called residually finite if the intersection of all its finite index
subgroups is trivial.
Theorem 6.9. Any finitely generated residually finite group G is Hopfian.
  It is easy to see that the finitely generated hypothesis cannot be omitted.
Proof. Let 1 = g ∈ G be in the kernel of an epimorphism φ from G to itself.
Let n > 0. There exist only finitely many subgroups of index n since G is
finitely generated. Pulling these subgroups back by φ, we get the same list
of index n subgroups, all of which contain g. It follows that g is in their
intersection. If G is residually finite, we obtain a contradiction.

 7. Unsolvability of certain decision problems in group theory
   The first goal of this section is to construct an explicit example of a group
which has an unsolvable word problem. What this means precisely is that
we have a group
                                  G= S|R
such that no algorithm can take an element of F (S) and determine whether
or not it represents the trivial word in G. An algorithm, on the other hand,
is a set of instructions which can be executed by a Turing machine. A Turing
machine consists of:
       • A finite set Q of states.
       • A finite alphabet Γ.
       • A distinguished blank symbol b ∈ Γ.
       • A transition function Q×Γ → Q×Γ×{L, R} which may be partially
       • An initial state q0 ∈ Q.
       • A set F ⊂ Q of final states.
   Less obscurely, a Turing machine takes an infinite tape which has all but
finitely many places marked with b, and finitely many symbols in Γ. The
Turing machine starts reading the tape in the initial state. After reading an
entry, the machine can change the entry on the tape, change its state, and
move the reader to the left or to the right.
   To say that a group G has a solvable word problem means that we give a
Turing machine a tape which has a word in the free group on the generators
written on it, and the machine tells us if the word was trivial or not. For
a free group on a finite set X, a Turing machine looks roughly as follows:
the alphabet consists of X ∪ X −1 ∪ b. We write a word on a tape without
inserting any blanks and feed it to the machine. If there are only blanks,
the machine moves to a terminal state which declares the word to be trivial.
Otherwise the machine moves to one of 2|X| states labeled by the elements
16                                T. KOBERDA

of X and X −1 . Every time the machine reads a generator, it moves to the
state labeled by the generator unless it is in a state qx and reads x−1 . If this
does not happen, the machine declares the word nontrivial. Otherwise, the
machine moves into a reduction loop. It replaces xx−1 by two blank spots
and then moves the letters it has not read over two spots until it reaches
another blank spot, which indicates that the word is over. The machine
then returns to the initial state and starts over.
   Let A be a countable subset of N. We say that A is computable if there
is a Turing machine which decides if n ∈ A or n ∈ A. We say that A is
computably enumerable or c.e. if there is a surjective computable function
f : N → A. A set is computable if and only if it and its complement are
c.e. The reader not familiar with this fact should check it. There is an easy
corollary to this observation:
Corollary 7.1. A finitely presented residually finite group G has a solvable
word problem.
Proof. Since G is residually finite, we can computably enumerate all el-
ements of G which are not the identity by enumerating homomorphisms
G → Sn as n → ∞. Since G is finitely presented, we can effectively enumer-
ate which elements represent the identity. Thus the set of elements which
represent the identity and those which do not are both c.e.

  Since there are only countably many computable sets (why?) and un-
countably many subsets of N, there exist noncomputable subsets of N.
Theorem 7.2. There exists a finitely generated group with an unsolvable
word problem.
Proof. Let S be a noncomputable subset of N. Let
                  G = a, b, c, d | a−i bai = c−i dci for i ∈ S .
Then solving the word problem in G allows us to compute S.

  A group is called computably presented if the set of relations is a c.e.
subset of F (S). The following is a difficult theorem of Higman:
Theorem 7.3. A group G embeds in a finitely presented group H if and
only if it is computably presented.
  It follows that if there exists a c.e., noncomputable subset of N, then there
exists a finitely presented group with an unsolvable word problem.
Lemma 7.4. There exists a c.e. set which is not computable.
Sketch. There exists a universal Turing machine F : N × N → N which
encodes all other Turing machines. Precisely, we can uniquely and effectively
code every Turing machine as a natural number. One way to do this is to
associate to each symbol or instruction an unambiguous number to get a
tuple (n1 , . . . , nk ) associated to a Turing machine Te . We then enumerate
the primes in increasing order and associate to Te the number
                                e = pn1 · · · pek .
                                     1         k
                                 COURSE NOTES                                    17

Let K ⊂ N × N be the set of pairs (e, n) such that Te converges on the
input n. Then K is c.e. by definition. K is not computable by a standard
diagonalization argument.
Proposition 7.5. There exist subgroups of Fn × Fn with unsolvable mem-
bership problems for n sufficiently large.
Proof. Let
                          H = x1 , . . . , xn | r1 , . . . , rm
be a group with an unsolvable word problem. Let LH < Fn ×Fn be generated
by elements (xi , xi ) and (1, rj ) for all i and j. It is an easy exercise to show
that (u, v) ∈ LH if and only if u = v in H.
   With a little more work, it is possible to show that there is no algorithm
which determines if a particular finite set of elements of Fn × Fn generates
a proper subgroup.
   The membership and generator problem for Fn × Fn have been of interest
historically because there are statements concerning maps from π1 (Σg ) →
Fg × Fg which are equivalent to the Poincar´ conjecture.
 Department of Mathematics, Harvard University, 1 Oxford St., Cambridge,
MA 02138
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