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The Coarse Geometry of Groups
Tim Susse
CUNY Graduate Center
December 2, 2011
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
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Outline
1
Groups as Geometric Objects
2
Quasi-isometries
3
Hyperbolicity and Hyperbolic Groups
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The Cayley Graph
Given a presentation of a group G = hS | Ri we associate a geometric
object called the Cayley graph, denoted Cayley(G, S).
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The Cayley Graph
Given a presentation of a group G = hS | Ri we associate a geometric
object called the Cayley graph, denoted Cayley(G, S).
Vertex Set = G
g1 ∼ g2 if and only if there exists s ∈ S ∪ S −1 with g1 = g2 s
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The Cayley Graph
Given a presentation of a group G = hS | Ri we associate a geometric
object called the Cayley graph, denoted Cayley(G, S).
Vertex Set = G
g1 ∼ g2 if and only if there exists s ∈ S ∪ S −1 with g1 = g2 s
Thus the Cayley graph of a group depends on the choice of generating
set.
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Cayley Graph of Z2
Figure: Cayley(Z2 , {(1, 0), (0, 1)})
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Cayley Graphs of Z/7Z
(a)
(b)
Figure: (a) Cayley(Z/7Z, {1}) (b) Cayley(Z/7Z, {2, 3})
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Two Cayley Graphs of Z
(a)
(b)
Figure: (a) Cayley(Z, {1}) (b) Cayley(Z, {2, 3})
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Two Cayley Graphs of Z
(a)
(b)
Figure: (a) Cayley(Z, {1}) (b) Cayley(Z, {2, 3})
You might notice that if you step really far back, or zoom out on the
graphic, the two Cayley graphs look very similar. We want to formalize
and take advantage of this.
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Word Metric on a Group
Given a generating set S for G, we define the distance between two
points in G to be their distance in the Cayley graph Cayley(G, S).
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The Coarse Geometry of Groups
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Word Metric on a Group
Given a generating set S for G, we define the distance between two
points in G to be their distance in the Cayley graph Cayley(G, S).
Again, this depends on the choice of generating set. This metric dS is
called the word metric on G.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
7 / 38
Word Metric on a Group
Given a generating set S for G, we define the distance between two
points in G to be their distance in the Cayley graph Cayley(G, S).
Again, this depends on the choice of generating set. This metric dS is
called the word metric on G.
Facts
For any g ∈ G, dS (g, e) = lS (g), the word length of G.
For any g, h ∈ G, dS (g, h) = lS (g −1 h).
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
7 / 38
Word Metric on a Group
Given a generating set S for G, we define the distance between two
points in G to be their distance in the Cayley graph Cayley(G, S).
Again, this depends on the choice of generating set. This metric dS is
called the word metric on G.
Facts
For any g ∈ G, dS (g, e) = lS (g), the word length of G.
For any g, h ∈ G, dS (g, h) = lS (g −1 h).
G acts on (G, dS ) by left multiplication, an isometry.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
7 / 38
Word Metric on a Group
Given a generating set S for G, we define the distance between two
points in G to be their distance in the Cayley graph Cayley(G, S).
Again, this depends on the choice of generating set. This metric dS is
called the word metric on G.
Facts
For any g ∈ G, dS (g, e) = lS (g), the word length of G.
For any g, h ∈ G, dS (g, h) = lS (g −1 h).
G acts on (G, dS ) by left multiplication, an isometry.
If S is a finite generating set, closed balls are finite, so (G, dS ) is a
proper metric space. The converse is also true.
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The Coarse Geometry of Groups
December 2, 2011
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Outline
1
Groups as Geometric Objects
2
Quasi-isometries
3
Hyperbolicity and Hyperbolic Groups
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Bilipschitz Equivalence
Given two metric spaces (X , dX ) and (Y , dY ) we say that a map
f : X → Y is a k -bilipschitz map if for any pair of points x1 , x2 ∈ X we
have:
1
dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ).
k
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Bilipschitz Equivalence
Given two metric spaces (X , dX ) and (Y , dY ) we say that a map
f : X → Y is a k -bilipschitz map if for any pair of points x1 , x2 ∈ X we
have:
1
dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ).
k
A bilipschitz map is like a stretching of the metric at every point (by
bounded amounts). It is worth noting the a bilipschitz map is always a
topological embedding.
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The Coarse Geometry of Groups
December 2, 2011
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Bilipschitz Equivalence
Given two metric spaces (X , dX ) and (Y , dY ) we say that a map
f : X → Y is a k -bilipschitz map if for any pair of points x1 , x2 ∈ X we
have:
1
dX (x1 , x2 ) ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ).
k
A bilipschitz map is like a stretching of the metric at every point (by
bounded amounts). It is worth noting the a bilipschitz map is always a
topological embedding.
A weaker notion of equivalence is a ”large scale” or coarse bilipschitz
condition. We call this a quasi-isometry.
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Quasi-isometric embeddings
Definition. We say that a map f : X → Y is a (k , c) quasi-isometric
embedding if for every pair x1 , x2 we have that
1
dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ) + c.
k
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Quasi-isometric embeddings
Definition. We say that a map f : X → Y is a (k , c) quasi-isometric
embedding if for every pair x1 , x2 we have that
1
dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) ≤ kdX (x1 , x2 ) + c.
k
Further, if the map is coarsely onto (i.e. if every point in Y is distance
at most c from f (X )), we call it a quasi-isometry.
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Quasi-isometric embeddings
Quasi-isometry defines an equivalence relation on metric spaces:
Clearly the identity map is an isometry, so any space is
quasi-isometric to itself.
A quick computation shows that if f and g are quasi-isometries, so
is f ◦ g
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Quasi-isometric embeddings
Quasi-isometry defines an equivalence relation on metric spaces:
Clearly the identity map is an isometry, so any space is
quasi-isometric to itself.
A quick computation shows that if f and g are quasi-isometries, so
is f ◦ g
Symmetry is a little tricky.
To show symmetry we need to construct a ”coarse inverse” of a
quasi-isometry f . By this we mean a quasi-isometry f −1 so that there
exists a constant r with dX (x, f −1 ◦ f (x)) ≤ r .
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The Coarse Geometry of Groups
December 2, 2011
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Coarse Inverse of a Quasi-isometry
To construct the inverse, we first need some quick facts.
If f : X → Y is a (k, c) quasi-isometry, then it’s failure to be injective is
bounded. In particular, if f (x1 ) = f (x2 ), then dX (x1 , x2 ) ≤ kc. To see
this, look at the left-side of the definition of quasi-isometry:
1
dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) = 0.
k
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The Coarse Geometry of Groups
December 2, 2011
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Coarse Inverse of a Quasi-isometry
To construct the inverse, we first need some quick facts.
If f : X → Y is a (k, c) quasi-isometry, then it’s failure to be injective is
bounded. In particular, if f (x1 ) = f (x2 ), then dX (x1 , x2 ) ≤ kc. To see
this, look at the left-side of the definition of quasi-isometry:
1
dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) = 0.
k
So, up to a bounded diameter ”error”, for each y ∈ f (X ), we can
choose g(y ) = x, where we choose some element x ∈ f −1 (y).
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
12 / 38
Coarse Inverse of a Quasi-isometry
To construct the inverse, we first need some quick facts.
If f : X → Y is a (k, c) quasi-isometry, then it’s failure to be injective is
bounded. In particular, if f (x1 ) = f (x2 ), then dX (x1 , x2 ) ≤ kc. To see
this, look at the left-side of the definition of quasi-isometry:
1
dX (x1 , x2 ) − c ≤ dY (f (x1 ), f (x2 )) = 0.
k
So, up to a bounded diameter ”error”, for each y ∈ f (X ), we can
choose g(y ) = x, where we choose some element x ∈ f −1 (y).
For each element y 6∈ f (X ), we note that there exists some y 0 ∈ f (X )
with dY (y, y 0 ) ≤ c. Choose some y 0 with this property and let g(y) = x
where x ∈ f −1 (y ).
This function is well-defined if we ignore these finite diameter
”errors”. We call this sort of thing coarsely well-defined.
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The Coarse Geometry of Groups
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Why is Quasi-isometry Right?
Given a finitely generated group G, there exist many finite generating
sets.
Proposition.
Let S and T be two finite generating sets of a group G, then the identity
map on G is a quasi-isometry idG : Cayley(G, S) → Cayley(G, T ).
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The Coarse Geometry of Groups
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Why is Quasi-isometry Right?
Given a finitely generated group G, there exist many finite generating
sets.
Proposition.
Let S and T be two finite generating sets of a group G, then the identity
map on G is a quasi-isometry idG : Cayley(G, S) → Cayley(G, T ).
Proof. Let S = {s1 , . . . , sn } and T = {t1 , . . . tm }. Since S generates G
there exists a shortest (geodesic) spelling of each ti in the language of
1
S. In particular, ti = s1i
· · · skik , where j = ±1. Let
K = max{k : lS (ti ) = k}, i.e. the longest word length of a ti in
Cayley(G, S). Similarly, let L be the maximum word length of the si in
Cayley(G, T ).
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Why is Quasi-isometry Right?
Take g ∈ G. Then say lS (g) = r . so g = s1 (g) · · · sr (g), where each
si (g) ∈ S ∪ S −1 . Replace each of the si with their spellings in t, and we
see that lT (g) ≤ Kr .
Similarly, if lT (g) = r , then lS (g) ≤ Lr . So, for any g ∈ G we get:
1
· lT (g) ≤ lS (g) ≤ L · lT (g).
K
However, for any g, h ∈ G, dS (g, h) = lS (gh−1 ), and similarly for T . So,
this turns in to a quasi-isometry. In fact, this is a bi-Lipschitz
equivalence.
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Why is Quasi-isometry Right?
Take g ∈ G. Then say lS (g) = r . so g = s1 (g) · · · sr (g), where each
si (g) ∈ S ∪ S −1 . Replace each of the si with their spellings in t, and we
see that lT (g) ≤ Kr .
Similarly, if lT (g) = r , then lS (g) ≤ Lr . So, for any g ∈ G we get:
1
· lT (g) ≤ lS (g) ≤ L · lT (g).
K
However, for any g, h ∈ G, dS (g, h) = lS (gh−1 ), and similarly for T . So,
this turns in to a quasi-isometry. In fact, this is a bi-Lipschitz
equivalence.
Notice in our Cayley graphs for Z before we had K = 3, L = 2 for
S = {1} and T = {2, 3}.
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The Coarse Geometry of Groups
December 2, 2011
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The Milnor-Svarc Lemma
One of the fundamental tools in studying the coarse geometry of
groups is the following fact.
Theorem.
Let X be a proper metric space and let G act on X geometrically
(properly discontinuously and cocompactly by isometries). Then G is
finitely generated and for a fixed x ∈ X the orbit map g 7→ gx is a
quasi-isometry.
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The Coarse Geometry of Groups
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The Milnor-Svarc Lemma
One of the fundamental tools in studying the coarse geometry of
groups is the following fact.
Theorem.
Let X be a proper metric space and let G act on X geometrically
(properly discontinuously and cocompactly by isometries). Then G is
finitely generated and for a fixed x ∈ X the orbit map g 7→ gx is a
quasi-isometry.
The proof involves taking a closed ball K (which is compact, since X is
proper) that contains a fundamental domain for the action on X . Let
S = {s ∈ G : K ∩ sK 6= ∅}. Since the action is properly discontinuous,
this set is finite.
We would then show that G = hSi by making a path from x ∈ K to
gx ∈ gK for g ∈ G by making steps that are small enough that the
corresponding translates of K intersect. These correspond to elements
of S. In doing this, the QI bounds fall out.
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The Milnor-Svarc Lemma
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Consequences of the Milnor-Svarc Lemma
If G is a finitely generated group and H ≤ G with [G : H] < ∞,
then H y G by multiplication on the right. This action is
cocompact, so H is QI to G. This is an example of what is called
commensurability (which is stronger than QI).
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Consequences of the Milnor-Svarc Lemma
If G is a finitely generated group and H ≤ G with [G : H] < ∞,
then H y G by multiplication on the right. This action is
cocompact, so H is QI to G. This is an example of what is called
commensurability (which is stronger than QI).
If G = π1 (M) where M is a compact Riemannian manifold, then G
e For instance
is finitely generated and is quasi-isometric to M.
π1 (S), a closed hyperbolic surface group (χ(S) < 0), is
quasi-isometric to the hyperbolic plane.
In a similar vein, Zn is quasi-isometric to Rn .
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QI and Algebra
There are many algebraic properties that have geometric content (i.e.
are quasi-isometry invariant). In particular:
Having a finite presentation is quasi-isometry invariant
Having a finite index free subgroup is a quasi-isometry invariant
Having two topological ends is equivalent to being virtually Z, so it
is a quasi-isometry invariant. Having one end is also a QI
invariant.
Having a finite index nilpotent subgroup (virtual nilpotence) is
equivalent to having polynomial growth (this is Gromov’s
Polynomial Growth Theorem). The latter is a quasi-isometry
invariant for groups.
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Outline
1
Groups as Geometric Objects
2
Quasi-isometries
3
Hyperbolicity and Hyperbolic Groups
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Slim Triangle Property
Defintion We say that a metric space (X , d) is δ-hyperbolic if for any
geodesic triangle [a, b, c] we have that d([a, b], [a, c] ∪ [b, c]) ≤ δ and
similarly for all other permutations of the letters. We call such a triangle
δ-thin or δ-slim
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Other Characterizations of Hyperbolicity
Hyperbolicity is innately a statement about ”large” triangles in a metric
space. In fact, any compact metric space is automatically hyperbolic
with δ equal to the (finite) diameter.
There are other, equivalent, notions of hyperbolicity.
Rips proved that a space is δ-hyperbolic if and only if every
quadruple of points a, b, p ∈ X satisfy
hb | cip ≥ min{ha | bip , ha | cip } − δ.
1
hx | y ip = (d(x, p) + d(y , p) − d(x, y)) is called the Gromov
2
Product.
Bowditch proved in his paper on the curve complex that
hyperbolicity can be formulated as a statement about coarse
centers of triangles [Bo].
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Examples of Hyperbolic Metric Spaces
As the name suggests, the hyperbolic plane H, which we identify
with the complex upper half plan with the Riemannian metric
√
dx 2 + dy 2
ds2 =
is log(1 + 2)-hyperbolic. Further, hyperbolic
2
y
space in any dimension is hyperbolic.
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Examples of Hyperbolic Metric Spaces
As the name suggests, the hyperbolic plane H, which we identify
with the complex upper half plan with the Riemannian metric
√
dx 2 + dy 2
ds2 =
is log(1 + 2)-hyperbolic. Further, hyperbolic
2
y
space in any dimension is hyperbolic.
A tree (of any kind, including an R-tree) is trivially 0-hyperbolic,
since any triangle is actually a tripod.
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Quasi-isometric Invariance
Proposition.
If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is also
hyperbolic.
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Quasi-isometric Invariance
Proposition.
If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is also
hyperbolic.
To prove this proposition, we need to study the properties of
quasi-isometric images of geodesics.
A geodesic is an isometric embedding of an interval.
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Quasi-isometric Invariance
Proposition.
If f : X → Y is a quasi-isometry and X is δ-hyperbolic, then Y is also
hyperbolic.
To prove this proposition, we need to study the properties of
quasi-isometric images of geodesics.
A geodesic is an isometric embedding of an interval.
A (k, c)-quasigeodesic is a (k , c) quasi-isometric embedding of
an interval.
Remark. If γ is a geodesic in X and f : X → Y is a
(k, c)-quasi-isometric embedding, then f (γ) is a
(k, c)-quasigeodesic.
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Quasi-isometric Invariance
Definition. A function e : N → R is called a divergence function for a
metric (length) space X if for every R, r ∈ N and any pair of geodesics
γ : [0, a] → X and γ 0 : [0, a0 ] → X with γ(0) = γ 0 (0) = x,
R + r ≤ min{a, a0 } and d(γ(R), γ 0 (R) ≥ e(0), any path connecting
γ(R + r ) to γ 0 (R + r ) outside B(x, R + r ) must have length atleast e(r ).
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Quasi-isometric Invariance
In the Euclidean plane we must have that e(r ) ≤ πr . Importantly,
any divergence function must be linear. As you might think,
divergence has to do with the size of a sphere of radius r .
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Quasi-isometric Invariance
In the Euclidean plane we must have that e(r ) ≤ πr . Importantly,
any divergence function must be linear. As you might think,
divergence has to do with the size of a sphere of radius r .
In an infinite tree, the divergence is infinite, since there is only one
path between two points. This is called a ”cut point”.
H2 has an exponential divergence function, this can be figured out
by computing the circumference of a circle of Euclidean radius r
centered at 0 in the disc model.
Is this also true in other hyperbolic spaces?
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Quasi-isometric Invariance
Theorem.
If X is a δ-hyperbolic metric space, then it has an exponential
divergence function.
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Quasi-isometric Invariance
Theorem.
If X is a δ-hyperbolic metric space, then it has an exponential
divergence function.
Proof.
Fix R, r ∈ N. Let γ and γ 0 be two geodesics based at some point x ∈ X
with d(γ(R), γ(R 0 )) > 2δ and set e(0) = 2δ.
Let p be a path in X \ B(x, R + r ) from γ(R + r ) to γ 0 (R + r ). and let α∅
be the geodesic from γ(R + r ) to γ 0 (R + r ). Now let m∅ be the middle
point on the path p and let α0 be the geodesic from γ(R + r ) to m∅ and
α1 the geodesic from m∅ to γ 0 (R + r ).
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Quasi-isometric Invariance
Theorem.
If X is a δ-hyperbolic metric space, then it has an exponential
divergence function.
Proof.
Now, for any binary string b, let mb be the midpoint of the segment of p
between the endpoints of αb . Now let αb0 be the geodesic between the
beginning of αb and mb and αb1 the geodesic between mb and the end
of αb . Keep subdividing p in this way until each segment in the division
has length between 12 and 1. If n is the number of pieces, then
log 2 l(p) ≤ n ≤ log 2 l(p) + 1.
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Quasi-isometric Invariance
Theorem.
If X is a δ-hyperbolic metric space, then it has an exponential
divergence function.
Proof.
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Quasi-isometric Invariance
Theorem.
If X is a δ-hyperbolic metric space, then it has an exponential
divergence function.
Proof.
For each b, the segments αb , αb0 , αb1 form a geodesic triangle, and so
are δ-slim.
Since d(γ(R), γ 0 (R)) > δ, there exists a point v (0) on α∅ with
d(v (0), γ(R)) < δ.
Continuing inductively, we can find v (1) on α0 ∪ α1 with
d(v (0), v (1)) ≤ δ. And so if v (i) is on αb we find v (i + 1) on either αb0
or αb1 with d(v (i), v (i + 1)) < δ.
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Quasi-isometric Invariance
Theorem.
If X is a δ-hyperbolic metric space, then it has an exponential
divergence function.
Proof.
Let v (m) be the point obtained at the last level of iteration. There is
apoint y ∈ P whose distance from v (m) is at most 1 and so its
distance from x is at most R + δ log2 (l(p)) + 2.
But d(x, P) ≥ R + r so
R + r ≤ R + δ log2 (l(p)) + 2,
i.e. l(p) is atleast exponential in r .
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Quasi-isometric Invariance
Proposition.
Let γ be a (k , c)-quasigeodesic with end points x and y and let [x, y]
denote a geodesic (not necessarily unique) connecting x to y . Then,
there exists M = M(k , c, δ) so that the Hausdorff distance between γ
and [x, y] is less than M.
In particular, γ is in the M-neighborhood of the geodesic between its
endpoints
The proof of this is an application of the theorem from the previous
slide.
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Quasi-isometric Invariance
Corollary.
If X is a δ-hyperbolic space and f : Y → X be a (K , C)-quasi-isometry,
then Y is K (2M + δ) + C-hyperbolic.
Proof.
Let [a, b, c] be some geodesic triangle in Y and consider its image
f ([a, b, c]) in X . Well f ([a, b]) is a quasi-geodesic, so it is in the
M-neighborhood of a geodesic [f (a), f (b)] and similarly for [b, c] and
[a, c]. Take a point x ∈ f ([a, b]). Let y ∈ [f (a), f (b)] be such that
d(x, y ) < M and let z ∈ [f (b), f (c)] ∪ [f (a), f (c)] be such that
d(y, z) < δ. Furthermore, there exists w ∈ f ([b, c]) ∪ f ([a, c]) such that
d(z, w) < M. So, d(x, w) < 2M + δ.
Since f was a quasi-isometry, and x and w are in the image of f , their
preimages on the triangle [a, b, c] are at most K (2M + δ) + C apart.
Thus the triangle in Y is slim.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
28 / 38
Quasi-isometric Invariance
Thanks to Victor Reyes for this image.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
29 / 38
Hyperbolic Groups
The previous slide shows us that it makes sense to talk about finitely
generated hyperbolic groups. If we transition between finite
generating sets, the Cayley graphs are quasi-isometric, so if one is
hyperbolic, so is any other.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
30 / 38
Hyperbolic Groups
The previous slide shows us that it makes sense to talk about finitely
generated hyperbolic groups. If we transition between finite
generating sets, the Cayley graphs are quasi-isometric, so if one is
hyperbolic, so is any other.
e is isometric to Hn ),
If M is a closed hyperbolic manifold (i.e. M
then π1 (M) is δ-hyperbolic. (This is a consequence of the
Milnor-Svarc Lemma)
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
30 / 38
Hyperbolic Groups
The previous slide shows us that it makes sense to talk about finitely
generated hyperbolic groups. If we transition between finite
generating sets, the Cayley graphs are quasi-isometric, so if one is
hyperbolic, so is any other.
e is isometric to Hn ),
If M is a closed hyperbolic manifold (i.e. M
then π1 (M) is δ-hyperbolic. (This is a consequence of the
Milnor-Svarc Lemma)
Given a hyperbolic group G, a subgroup H is quasiconvex if and
only if the inclusion map H ,→ G is a quasi-isometric embedding.
In a hyperbolic group, if g ∈ G is infinite order, then the centralizer
of g is quasiconvex.
No hyperbolic group contains as Z2 subgroup.
In fact, it can
not
contain a Baumslag-Solitar subgroup a, b | b−1 am b = an .
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
30 / 38
Hyperbolic Groups
e is isometric to Hn ),
If M is a closed hyperbolic manifold (i.e. M
then π1 (M) is δ-hyperbolic. (This is a consequence of the
Milnor-Svarc Lemma)
Given a hyperbolic group G, a subgroup H is quasiconvex if and
only if the inclusion map H ,→ G is a quasi-isometric embedding.
In a hyperbolic group, if g ∈ G is infinite order, then the centralizer
of g is quasiconvex.
No hyperbolic group contains as Z2 subgroup.
In fact, it can
not
contain a Baumslag-Solitar subgroup a, b | b−1 am b = an .
[Gromov] If G satisfies the small cancellation condition C 0 ( 61 ), then
G is hyperbolic.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
30 / 38
Dehn Presentations
It’s natural to wonder if the word problem is solvable in a hyperbolic
group (or any new class of groups that is defined). It turns out that
hyperbolic groups have very special presentations that make the word
problem easy.
Definition. Let G = hS | Ri. We say that the presentation is a Dehn
presentation if for any reduced word w with w = 1 in G, there exists a
relator r ∈ R so that r = r1 r2 , l(r1 ) > l(r2 ) and w = w1 r1 w2 .
In other words, any word that represents the identity in G contains
more than one half of a relator, and so it can be shortened.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
31 / 38
Dehn Presentations
It’s natural to wonder if the word problem is solvable in a hyperbolic
group (or any new class of groups that is defined). It turns out that
hyperbolic groups have very special presentations that make the word
problem easy.
Definition. Let G = hS | Ri. We say that the presentation is a Dehn
presentation if for any reduced word w with w = 1 in G, there exists a
relator r ∈ R so that r = r1 r2 , l(r1 ) > l(r2 ) and w = w1 r1 w2 .
In other words, any word that represents the identity in G contains
more than one half of a relator, and so it can be shortened.
A word which cannot be further reduced or shortened by this method
(replacing r1 by r2−1 , a shorter word) is called Dehn reduced.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
31 / 38
Efficient Solution to the Word Problem
If G has a finite Dehn presentation (G is finitely generated, R is finite)
then you can check all subwords of length at most
N = max {l(r ) : r ∈ R} to see if a reduction can be made.
This procedure for solving the word problem is called Dehn’s
Algorithm, originally created by Max Dehn in 1910 to solve the word
problem in surface groups. Its run time is O(|w|2 ) in its simplest
iteration. (There are atmost |w| − N subwords in each step, and at
most |w| steps in the reduction.)
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
32 / 38
Efficient Solution to the Word Problem
If G has a finite Dehn presentation (G is finitely generated, R is finite)
then you can check all subwords of length at most
N = max {l(r ) : r ∈ R} to see if a reduction can be made.
This procedure for solving the word problem is called Dehn’s
Algorithm, originally created by Max Dehn in 1910 to solve the word
problem in surface groups. Its run time is O(|w|2 ) in its simplest
iteration. (There are atmost |w| − N subwords in each step, and at
most |w| steps in the reduction.)
Example. G = ha, b, c, d | [a, b][c, d]i. Let R be the symmetrized set of
generators, so that R contains all cyclic conjugates of [a, b][c, d] and
its inverse. What about other hyperbolic groups?
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
32 / 38
Hyperbolic Groups: Local Geodesics
Definition. A path γ in a metric space X is called a k-local geodesic if
every subpath of length k is a geodesic.
On a sphere of radius 1, a great circle is a π-local geodesic.
If M is a Riemannian manifold, let r (M) be the injectivity radius of
M. Then any image of a ray in Tx M under the exponential map is
an r (M)-local geodesic.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
33 / 38
Hyperbolic Groups: Local Geodesics
Definition. A path γ in a metric space X is called a k-local geodesic if
every subpath of length k is a geodesic.
On a sphere of radius 1, a great circle is a π-local geodesic.
If M is a Riemannian manifold, let r (M) be the injectivity radius of
M. Then any image of a ray in Tx M under the exponential map is
an r (M)-local geodesic.
Local geodesics have to do with loops in a metric space, or relations in
a Cayley graph. That makes them natural to study when considering
presentations.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
33 / 38
Hyperbolic Groups: Local Geodesics
Lemma.
Let G be δ-hyperbolic group and let γ be a 4δ-local geodesic. Let g be
the geodesic between the endpoints of γ (called γ+ and γ− ). Assume
l(g) > 2δ and let r and s be points on γ and g respectively, both
distance 2δ from γ+ . Then d(r , s) ≤ δ.
The proof here is by induction on the length of γ and uses the thin
triangle property multiple times.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
34 / 38
Hyperbolic Groups: Dehn Presentation
Theorem.
If γ is a 4δ-local geodesic in a δ-hyperbolic group G, then γ is
contained in the 3δ neighborhood of the geodesic between its
endpoints.
We will use this theorem to create a shortening algorithm in our
hyperbolic group G. Fix a finite generating set S. Let
R = {w : w = 1 in G, |w| < 8δ}. We aim to show that hS | Ri is a Dehn
presentation for G.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
35 / 38
Hyperbolic Groups: Dehn Presentation
With R as in the previous slide, take a word w in the generators so that
w = 1 in G. If the loop w in the Cayley graph is already a 4δ-local
geodesic, then it is in the 3δ neighborhood of the origin, so it is already
an element of R (since if it had length more than 8δ, the first 4δ long
segment of the geodesic would leave the ball of radius 3δ) and we can
see that it represents the identity.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
36 / 38
Hyperbolic Groups: Dehn Presentation
With R as in the previous slide, take a word w in the generators so that
w = 1 in G. If the loop w in the Cayley graph is already a 4δ-local
geodesic, then it is in the 3δ neighborhood of the origin, so it is already
an element of R (since if it had length more than 8δ, the first 4δ long
segment of the geodesic would leave the ball of radius 3δ) and we can
see that it represents the identity.
Now say that w is not a 4δ-local geodesic. Then there exists some
subpath (subword) w1 of length 4δ which is not a geodesic between its
endpoints. Replace it in w by the geodesic between its endpoints, call
it w2 . Then the path w1 w2−1 has length less than 8δ, so that word is in
R. Further, we note that w contained w1 , the longer part of the relator.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
36 / 38
Hyperbolic Groups: Dehn Presentation
With R as in the previous slide, take a word w in the generators so that
w = 1 in G. If the loop w in the Cayley graph is already a 4δ-local
geodesic, then it is in the 3δ neighborhood of the origin, so it is already
an element of R (since if it had length more than 8δ, the first 4δ long
segment of the geodesic would leave the ball of radius 3δ) and we can
see that it represents the identity.
Now say that w is not a 4δ-local geodesic. Then there exists some
subpath (subword) w1 of length 4δ which is not a geodesic between its
endpoints. Replace it in w by the geodesic between its endpoints, call
it w2 . Then the path w1 w2−1 has length less than 8δ, so that word is in
R. Further, we note that w contained w1 , the longer part of the relator.
Now, we can continue this process until we reduce w to a 4δ-local
geodesic, a case we have already covered. Thus, G = hS | Ri is a
Dehn presentation.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
36 / 38
Open Questions [Be]
[Gromov] Given a hyperbolic group G with one topological end
(i.e. a freely indecomposable hyperbolic group), does it contain a
surface subgroup?
Are hyperbolic groups residually finite?
[Bestvina] Say that G admits a finite dimensional K (G, 1) and
does not contain any Baumslag-Solitar groups. Is G necessarily
hyperbolic? If G embeds in a hyperbolic group is this true? (Note:
Gromov proved that every hyperbolic group admits a finite
dimensional K(G, 1), making this question more natural than it
seems.)
[Canary] Let H ≤ G, G a hyperbolic group. If there exists some n
so that g n ∈ H for every g ∈ G, is H necessarily finite index in G?
(The answer is yes is H is quasiconvex)
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
37 / 38
References
[Be] Mladen Bestvina, Questions in Geometric Group Theory,
http://www.math.utah.edu/ bestvina/eprints/questions-updated.pdf
(2004).
[Bo] Brian Bowditch, Intersetction Numbers and Hyperbolicity of the
Curve Complex, J. reine angew. Math. 598 (2008), 105-129.
[BrH] Martin Bridson and Andre Häfliger, Metric Spaces of
Non-positive Curvature: Grundlehren der mathematischen
Wissenschaften Series, Springer (2010).
[Gr] M. Gromov, Hyperbolic Groups: Essays in Group Theory, S. M.
Gersten ed., M.S.R.I. Publ 8, Springer (1988), 75 - 263.
Tim Susse CUNY Graduate Center ()
The Coarse Geometry of Groups
December 2, 2011
38 / 38