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Introduction to Hyperbolic Metric Spaces

Adriana-Stefania Ciupeanu

University of ManitobaDepartment of Mathematics

November 3, 2017

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Introduction

Introduction

Geometric group theory is relatively new, and became a clearlyidentifiable branch of mathematics in the 1990s due to Mikhail Gromov.

Hyperbolicity is a centre theme and continues to drive current researchin the field.

Geometric group theory is bases on the principle that if a group acts assymmetries of some geometric object, then we can use geometry tounderstand the group.

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Introduction

Introduction

Gromov’s notion of hyperbolic spaces and hyperbolic groups havebeen studied extensively since that time.

Many well-known groups, such as mapping class groups andfundamental groups of surfaces with cusps, do not meet Gromov’scriteria, but nonetheless display some hyperbolic behaviour.

In recent years, there has been interest in capturing and using thishyperbolic behaviour wherever and however it occurs.

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Introduction

If Euclidean geometry describes objects in a flat world or a plane, andspherical geometry describes objects on the sphere, what world doeshyperbolic geometry describe?

Hyperbolic geometry takes place on a curved two dimensional surfacecalled hyperbolic space.

The essential properties of the hyperbolic plane are abstracted toobtain the notion of a hyperbolic metric space, which is due to Gromov.

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Introduction

Hyperbolic geometry is a non-Euclidean geometry, where the parallelpostulate of Euclidean geometry is replaced with:

For any given line R and point P not on R, in the plane containing bothline R and point P there are at least two distinct lines through P that donot intersect R.

Because models of hyperbolic space are big (not to mention infinite),we will do all of our work with a map of hyperbolic space called thePoincaré disk.

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Introduction

The Poincaré disk

D = {(x , y) | x2 + y2 < 1}

is the inside of a circle (although the circle is not included) with metric

ds2 =dx2 + dy2

(1− x2 − y2)2

In the Poincaré disk model, geodesics appear curved. They are arcs ofcircles. Specifically:

Geodesics are arcs of circles which meet the edge of the disk atπ

2.

Geodesics which pass through the center of the disk appearstraight.

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Introduction

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Introduction

The essential properties of the hyperbolic plane are abstracted toobtain the notion of a hyperbolic metric space, which is due to Gromov.

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Geodesic

Geodesic Metric Spaces

DefinitionLet (X ,d) be a metric space and p : [0,1]→ X be a path. If thesupremum over all finite partitions [t0 = 0, t2, . . . , tn−1, tn = 1] of

n

∑i=1

d(p(ti−1, p(t1)),

exists then we say that p is a rectifiable path. We denote thesupremum by l(p) and call it the length of p. Furthermore, (X , d) is apath metric space if for all x1 and x2 in X , we have that

d(x1, x2) = inf{l(p) | p is a rectifiable path in X from x1 to x2}.

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Geodesic

Geodesic Metric Spaces

The idea is as follows. Suppose you want to formally define the notionof length of a path. One property you would like is that if youapproximate your curve by a concatenation of “straight lines” then thelength of such concatenation approximates the length of the path (well,if the path has finite length). The length of a “straight line” should bethe same as the distance between the endpoint.

DefinitionA metric space X is geodesic if between every two points of X thereexists a geodesic.

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Geodesic

Geodesic Metric Spaces

A geodesic metric space is a length space in which infima of distancesof rectifiable paths are attained. Such a space clearly has to be pathconnected.

ExampleA connected graph is a geodesic metric space. We give each edgelength and as above the distance between two vertices v1 and v2 is theleast number of edges of a path between v1 and v2.

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Geodesic

Example

Example

Consider the unit circle S1 in the plane. There are two natural metricswe could put on S1:

The first is the induced Euclidean metric: the distance betweentwo points is the length of the straight line in R2 between them.The other is the arc length metric: the distance between twopoints is the length of the (shortest) circular arc between them.

The first of these is not a geodesic metric (since, for example, there isno path is S1 of length 2 connecting a pair of antipodal points),whereas the second one is geodesic.

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Geodesic

Slim and Thin Triangles

We want to know what it means for a space (X , d) to be hyperbolic.There are several definitions, all of them being equivalent.

We will give various versions of Gromov’s hyperbolic criterion.

DefinitionSuppose that (X ,d) is a metric space with basepoint w. Then wedefine the Gromov product on X × X based at w by

(x · y)w =12(d(x ,w) + d(y ,w)− d(x , y)) .

This is sometimes called an inner product. However, it is not an innerproduct in the usual sense as it is not necessary defines on a vectorspace.

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Geodesic

Example

Suppose that x , y , and w are points in a tree T , let γ be a geodesicfrom w to x and let η be a geodesic from w to y . Then (x · y)w is thedistance along γ (or η) before the two geodesics separate.

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Geodesic

Example

Choose any basepoint w in T . Then the metric d on T satisfies theproperty that for all x , y , and z in T we have that

(x · z)w ≥ min{(x · y)w , (y · z)w}.

We have that there are three cases:

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Geodesic

Example

In case (a) we have

(x · z)w = (x · y)w = min{(x · y)w , (y · z)w},

in (b) we have(x · z)w > (x · y)w = (y · z)w ,

and in (c) we have

(x · z)w = (y · z)w = min{(x · y)w , (y · z)w}.

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Geodesic

Slim and Thin Triangles

If we relax the property satisfies by the metric on a tree in the aboveexample then we get the following:

DefinitionIf (X ,d) is a metric space with basepoint w and there exists δ ≥ 0such thta for all x , y , and z in X we have

(x · z)w ≥ min{(x · y)w , (y , z)w} − δ,

then we say that the Gromov product based at w is δ-hyperbolic. Ifthere exists δ ≥ 0 such that the Gromov product is δ-hyperbolic, wejust say that the Gromov product based at w is hyperbolic.

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Geodesic

Slim and Thin Triangles

We next consider geodesic triangles in a geodesic metric space, i.e.triples of points with a specified geodesic (called a side) between everypair of points. If (x , y , z) is such a triangle then we denote thegeodesic between, say, x and y by [x , y ].

DefinitionLet (X ,d) be a geodesic metric space and let 4 be a geodesictriangle. We say that 4 is δ-slim if for each ordering (A,B,C) of itssides, and for any point w ∈ A, w ∈ B, w ∈ B we have

min{(.w ,B),d(w ,C)} ≤ δ

min{(.w ,A),d(w ,B)} ≤ δ

min{(.w ,A),d(w ,B)} ≤ δ

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Geodesic

Slim and Thin Triangles

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Geodesic

Slim and Thin Triangles

In other words, we have that a δ− neighbourhood of any two sidescontains the third side.

Typical euclidean triangles are not all δ-slim for some fixed δ. If you fixδ, just choose a very large triangle.

DefinitionLet X be a geodesic metric space. Given a geodesic triangle δ in X ,we consider the Euclidean triangle 4′ with the same side lengths.Collapse 4′ to a tripod T . Let p : 4 → T be the map which so arises.We say that 4 is δ-thin if for all t ∈ T we have diam(p−1(t)) ≤ δ.

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Geodesic

Slim and Thin Triangles

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Hyperbolic Metric Spaces

Hyperbolic Metric Space

PropositionLet X be a geodesic metric space. Then the following are equivalent.

1 There exists a point w ∈ X such that the Gromov product basedat w is hyperbolic.

2 For all points w ∈ X the Gromov product based at w is hyperbolic.3 There exists δ ≥ 0 such that every geodesic triangle in X is δ-slim.4 There exists δ ≥ 0 such that every geodesic triangle in X is δ-thin.

DefinitionIf a geodesic metric space satisfies any of the properties in the lastproposition we call it a hyperbolic metric space.

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Hyperbolic Metric Spaces

Hyperbolic Metric Spaces

ExamplesAny tree is 0-hyperbolic. Every geodesic triangle is a “tripod“.

Example

R2 is not δ-hyperbolic for any δ.

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Hyperbolic Metric Spaces

Example Hyperbolic Plane

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Hyperbolic Metric Spaces

Example Hyperbolic Plane

The hyperbolic plane is hyperbolic.Incircle of a geodesic triangle is the circle of largest diametercontained in the triangleEvery geodesic triangle lies in the interior of an ideal triangle, all ofwhich are isometric with incircles of diameter 2 log 3.The Gromov product also has a simple interpretation in terms ofthe incircle of a geodesic triangle.The quantity (A,B)C is just the hyperbolic distance p from C toeither of the points of contact of the incircle with the adjacent

sides: c = (a− p) + (b− p), so that p =(a + b− c)

2= (A,B)C .

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Hyperbolic Metric Spaces

Cayley Graph

It can be showed that if two spaces are quasi-isomteric and one ofthem is hyperbolic, then so is the other (the constant δ might change).

In particular, we have that a finitely generated group G is hyperbolic ifand only the Cayley graph of G is hyperbolic. However, this is up toquasi-isometry.

Furthermore, we have that given a group G and different generatingsets the Cayles graphs are different.

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Hyperbolic Metric Spaces

Cayley Graph of Z/6Z with generating set {1}

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Hyperbolic Metric Spaces

Cayley Graph of Z/6Z with generating set {2, 3}

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Quasi-Isometry

Quasi-Isometries

DefinitionLet (X ,dX ) and (Y ,dY ) be two metric spaces. A map f : X → Y is a(λ, c) QI embedding if for all x , y ∈ X we have (λ ≥ 1, c ≥ 0)

dX (x , y)λ

− c ≤ dY (f (x), f (y)) ≤ λdx (x , y) + c.

Furthermore, the (λ, c) QI embedding is a (λ, c) QI is for every y ∈ Ythere exists x ∈ X where

dy f (x), y) ≤ c.

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Quasi-Isometry

Quasi-Isometries

ExampleIf X and Y are both bounded metric spaces, then X and Y arequasi-isometric.

ExampleAny finite radius metric space (X ,d) is QI to a point. Letλ = sup{d(x , y) | x , y ∈ X}. Let c = 1. Furthermore, let f : X → {p}be the constant map. We want to show that for every x , y ∈ X

d(x , y)λ

− c1 ≤ (f (x), f (y)) =≤ λd(x , y) + 1.

Since, is onto it is QI.

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Quasi-Isometry

Quasi-Isometry

Example

For v ,b ∈ R2, the map t 7→ tv + b from R to R2 is a quasi-isometricembedding.

Example

The map t 7→ t2 from R to R is not a quasi-isometric embedding. Thesecond inequality is the one that fails.

Example

The map t 7→√

t from R to R is not a quasi-isometric embedding. Thistime the first inequality fails.

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Quasi-Isometry

Quasi-Isometry

ExampleLet G be a group given as the quotient π : F (S)→ G and also givenas π′ : F (S′)→ G, where S and S′ are both finite sets. Now, supposethat d and d ′ are corresponding on G with respect to S and S′. Now,we have

1 : G→ G

is a QI between (G,d) to (G,d ′)Let λ1 = max{d ′(π(s),1) | s ∈ S}, λ2 = max{d ′(π′(s′),1) | s′ ∈ S′}and λ = max{λ1,λ2}. Then for all x , y ∈ G we have

dS(x , y)λ

≤ dS′(x , y) ≤ λdS(x , y)

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Quasi-Isometry

Quasi-Isometry

Example (Continuation)Let us prove that

dS(x , y)λ

≤ dS′(x , y).

Suppose that dS′(x , y) = n, for some n ∈ Z, then there existss′1, . . . , s′n such that

π(s′1, . . . , s′n) = x−1y =⇒ π(s′1) . . . π(s′n) = x−1y .

This will give us a walk of length λn between 1 to x1y in G consideredas the quotient of S. Therefore, dS(x , y) ≤ λn. Hence, we have that

dS(x , y)λ

≤ dS′(x , y).

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Quasi-Isometry

Quasi-Isometry

ExampleConsider the graph in the figure below. Collapsing each of the trianglesto a point gives a quasi-isometry of this graph onto a tree.

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Quasi-Isometry

Quasi-Isometry

Definition (Quasi-inverse)Let f : (X ,d)→ (Y ,d ′) be a map of metric spaces. We say thatg : (Y ,d ′)→ (X ,d) is a quasi-inverse of if there exists some D ≥ 0such that for all x ∈ X we have d(g ◦ f (x), x) ≤ D and for all y ∈ Y ehave d(f ◦ g(y), y) ≤ D.

PropositionComposition of two compatible quasi-isometries is again aquasi-isometry.

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Quasi-Isometry

Conclusion

Suppose that we have proved that a group G is hyperbolic. Thenwhat?

G is finitely generated.For g ∈ G the centralizer CG(g) is virtually cyclic, i.e., contains afinite index cyclic group.G has solvable word problem, i.e., if we are given an arbitraryword in the generators of G then there exists an algorithm todecide whether or not this word represents the identity.Same for the conjugacy and isomorphism problems.

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