4Hyperbolic groups

IV Topics in Geometric Group Theory 4.3 Dehn functions of hyperbolic groups
We now use our new understanding of quasi-geodesics in hyperbolic spaces to
try to understand the word problem in hyperbolic groups. Note that by the
Schwarz–Milnor lemma, hyperbolic groups are finitely-generated and their Cayley
graphs are hyperbolic.
Corollary.
Let
X
be
δ
-hyperbolic. Then there exists a constant
C
=
C
(
δ
) such
that any non-constant loop in X is not C-locally geodesic.
Proof. Take k = 8δ + 1, and let
C = max{λε, k}
where λ, ε are as in the theorem.
Let
γ
: [
a, b
]
X
be a closed loop. If
γ
were
C
-locally geodesic, then it
would be (λ, ε)-quasigeodesic. So
0 = d(γ(a), γ(b))
|b a|
λ
ε.
So
|b a| λε < C.
But γ is a C-local geodesic. This implies γ is a constant loop.
Definition
(Dehn presentation)
.
A finite presentation
hS | Ri
for a group Γ
is called Dehn if for every null-homotopic reduced word
w S
, there is (a
cyclic conjugate of) a relator
r R
such that
r
=
u
1
v
with
`
S
(
u
)
< `
S
(
v
), and
w = w
1
vw
2
(without cancellation).
some part in the word that can be replaced with a shorter word using a single
relator.
If a presentation is Dehn, then the naive way of solving the word problem
just works. In fact,
Lemma. If Γ has a Dehn presentation, then δ
Γ
is linear.
Proof. Exercise.
Theorem.
Every hyperbolic group Γ is finitely-presented and admits a Dehn
presentation.
In particular, the Dehn function is linear, and the word problem is solvable.
So while an arbitrary group can be very difficult, the generic group is easy.
Proof.
Let
S
be a finite generating set for Γ, and
δ
a constant of hyperbolicity
for Cay
S
(Γ).
Let C = C(δ) be such that every non-trivial loop is not C-locally geodesic.
Take
{u
i
}
to be the set of all words in
F
(
S
) representing geodesics [1
, u
i
] in
Cay
S
(Γ) with
|u
i
| < C
. Let
{v
j
} F
(
S
) be the set of all non-geodesic words of
length C in Cay
S
(Γ). Let R = {u
1
i
v
j
F (S) : u
i
=
Γ
v
j
}.
We now just observe that this gives the desired Dehn presentation! Indeed,
any non-trivial loop must contain one of the
v
j
’s, since
Cay
S
(Γ) is not
C
-locally
geodesic, and re can replace it with u
i
!
This argument was developed by Dehn to prove results about the fundamental
group of surface groups in 1912. In the 1980’s, Gromov noticed that Dehn’s
argument works for an arbitrary hyperbolic group!
One can keep on proving new things about hyperbolic groups if we wished
to, but there are better uses of our time. So for the remaining of the chapter,
we shall just write down random facts about hyperbolic groups without proof.
So hyperbolic groups have linear Dehn functions. In fact,
Theorem
(Gromov, Bowditch, etc)
.
If Γ is a finitely-presented group and
δ
Γ
ň n
2
, then Γ is hyperbolic.
Thus, there is a “gap” in the isoperimetric spectrum. We can collect our
results as
Theorem. If Γ is finitely-generated, then the following are equivalent:
(i) Γ is hyperbolic.
(ii) Γ has a Dehn presentation.
(iii) Γ satisfies a linear isoperimetric inequality.
(iv) Γ has a subquadratic isoperimetric inequality.
In general, we can ask the question for which
α R
is
n
α
'
a Dehn
function of a finitely-presented group? As we saw,
α
cannot lie in (1
,
2), and
it is a theorem that the set of such
α
is dense in [2
,
). In fact, it includes all
rationals in the interval.
Subgroup structure
When considering subgroups of a hyperbolic group Γ, it is natural to consider
“geometrically nice” subgroups, i.e. finitely-generated subgroups
H
Γ such
that the inclusion is a quasi-isometric embedding. Such subgroups are called
quasi-convex, and they are always hyperbolic.
What sort of such subgroups can we find? There are zillions of free quasi-
convex subgroups!
Lemma
(Ping-pong lemma)
.
Let Γ be hyperbolic and torsion-free (for conve-
nience of statement). If
γ
1
, γ
2
Γ do not commute, then for large enough
n
, the
subgroup hγ
n
1
, γ
n
2
i
=
F
2
and is quasi-convex.
How about non-free subgroups? Can we find surface groups? Of course, we
cannot always guarantee the existence of such surface groups, since all subgroups
of free groups are free.
Question.
Let Γ be hyperbolic and torsion-free, and not itself free. Must Γ
contain a quasi-convex subgroup isomorphic to
π
1
Σ for some closed hyperbolic
surface Σ?
We have no idea if it is true or false.
Another open problem we can ask is the following:
Question.
If Γ is hyperbolic and not the trivial group, must Γ have a proper
subgroup of finite index?
Proposition.
Let Γ be hyperbolic, and
γ
Γ. Then
C
(
γ
) is quasiconvex. In
particular, it is hyperbolic.
Corollary. Γ does not contain a copy of Z
2
.
The boundary
Recall that if Σ is a compact surface of genus
g
2, then
π
1
Σ
'
qi
H
2
. If we try
to draw the hyperbolic plane in the disc model, then we would probably draw a
circle and fill it in. One might think the drawing of the circle is just an artifact
of the choice of the model, but it’s not! It’s genuinely there.
Definition
(Geodesic ray)
.
Let
X
be a
δ
-hyperbolic geodesic metric space. A
geodesic ray is an isometric embedding r : [0, ) X.
We say
r
1
r
2
if there exists
M
such that
d
(
r
1
(
t
)
, r
2
(
t
))
M
for all
t
. In
the disc model of
H
2
, this is the scenario where two geodesic rays get very close
together as
t
. For example, in the upper half plane model of
H
2
, all vertical
lines are equivalent in this sense.
We define
X
=
{geodesic rays}/
. This can be topologized in a sensible
way, and in this case
X
X
is compact. By the Morse lemma, for hyperbolic
spaces, this is quasi-isometry invariant.
Example.
If Γ =
π
1
Σ, with Σ closed hyperbolic surface, then
Γ =
S
1
and
the union X
X gives us the closed unit disc.
Theorem
(Casson–Jungreis, Gabai)
.
If Γ is hyperbolic and
Γ
=
S
1
, then Γ
is virtually π
1
Σ for some closed hyperbolic Σ.
Example. If Γ is free, then
Γ is the Cantor set.
Conjecture
(Cannon)
.
If Γ is hyperbolic and
Γ
=
S
2
, then Γ is virtually
π
1
M for M a closed hyperbolic 3-manifold.