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Cornelia DRUT¸U

Universit´e de Lille I,

Cornelia.Drutu@math.univ-lille1.fr

Contents

1 Preliminaries on quasi-isometries 2

1.1 Basic deﬁnitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Examples of quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Rigidity of non-uniform rank one lattices 6

2.1 Theorems of Richard Schwartz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Finite volume real hyperbolic manifolds . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Proof of Theorem 2.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Classes of groups complete with respect to quasi-isometries 14

3.1 List of classes of groups q.i. complete. . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Relatively hyperbolic groups: preliminaries . . . . . . . . . . . . . . . . . . . . . 15

4 Asymptotic cones of a metric space 18

4.1 Deﬁnition, preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 A sample of what one can do using asymptotic cones . . . . . . . . . . . . . . . . 21

4.3 Examples of asymptotic cones of groups . . . . . . . . . . . . . . . . . . . . . . . 22

5 Relatively hyperbolic groups: image from inﬁnitely far away and rigidity 23

5.1 Tree-graded spaces and cut-points . . . . . . . . . . . . . . . . . . . . . . . . . . 23

5.2 The characterization of relatively hyperbolic groups in terms of asymptotic cones 25

5.3 Rigidity of relatively hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . 27

5.4 More rigidity of relatively hyperbolic groups: outer automorphisms . . . . . . . . 29

6 Groups asymptotically with(out) cut-points 30

6.1 Groups with elements of inﬁnite order in the center, not virtually cyclic . . . . . 31

6.2 Groups satisfying an identity, not virtually cyclic . . . . . . . . . . . . . . . . . . 31

6.3 Existence of cut-points in asymptotic cones and relative hyperbolicity . . . . . . 33

7 Open questions 34

8 Dictionary 35

1These notes represent a slightly modiﬁed version of the lectures given at the summer school

“G´eom´etries a` courbure n´egative ou nulle, groupes discrets et rigidite´s” held from the 14-th of

June till the 2-nd of July 2004 in Grenoble.

Many of the open questions formulated in the paper do not belong to the author and have

been asked by other people before.

1 Preliminaries on quasi-isometries

Nota bene: In order to ensure some coherence in the exposition, some notions are not deﬁned

in the text, but in a Dictionary at the end of the text.

1.1 Basic deﬁnitions

A quasi-isometric embedding of a metric space (X,dist ) into a metric space (Y,dist ) is a mapX Y

q :X!Y such that for every x ,x "X,1 2

1

dist (x ,x )#C$ dist (q(x ),q(x ))$Ldist (x ,x)+C, (1)X 1 2 Y 1 2 X 1 2L

for some constants L% 1 and C% 0.

IfX is a ﬁnite interval [a,b] then q is called quasi-geodesic (segment). If a =#& or b=+&

then q is called quasi-geodesic ray. If both a =#& and b=+& then q is called quasi-geodesic

line. The same names are used for the image of q.

If moreover Y is contained in theC–tubular neighbourhood of q(X) then q is called a quasi-

¯ ¯ ¯isometry. In this case there exists q : Y ! X quasi-isometry such that q' q and q' q are at

uniformly bounded distance from the respective identity maps (see [GH ] for a proof). We call2

¯q quasi-converse of q.

The objects of study are the ﬁnitely generated groups. We ﬁrst recall how to make them

into geometric objects. Given a group G with a ﬁnite set of generators S containing together

with every element its inverse, one can construct the Cayley graph Cayley(G,S) as follows:

• its set of vertices is G;

• every pair of elements g ,g "G such thatg =g s, with s"S, is joined by an edge. The1 2 1 2

oriented edge (g ,g ) is labeled by s.1 2

We supposethat every edge has length 1and we endow Cayley(G,S) with thelength metric.

Its restriction to G is called the word metric associated to S and it is denoted by dist . SeeS

Figure 1 for the Cayley graph of the free group of rank twoF =(a,b).2

Remark 1.1. A Cayley graph can be constructed also for an inﬁnite set of generators. In this

case the graph has inﬁnite valence in each point.

¯We note that if S and S are two ﬁnite generating sets of G then dist and dist are bi-¯S S

2Lipschitz equivalent. In Figure 2 are represented the Cayley graph ofZ with set of generators

2{(1,0),(0,1)} and the Cayley graph ofZ with set of generators {(1,0),(1,1)}.

2b

!1

a a

!1

b

Figure 1: Cayley graph ofF .2

2Figure 2: Cayley graph ofZ .

31.2 Examples of quasi-isometries

1. The main example, which partly justiﬁes the interest in quasi-isometries, is the following.

!Given M a compact Riemannian manifold, let M be its universal covering and let ! (M)1

be its fundamental group. The group ! (M) is ﬁnitely generated, in fact even ﬁnitely1

presented [BrH, Corollary I.8.11, p.137].

!The metric space M with the Riemannian metric is quasi-isometric to ! (M) with some1

word metric. This can be clearly seen in the case when M is the n-dimensional ﬂat torus

n n n n!T . In this caseM isR and ! (M) isZ . They are quasi-isometric, asR is a thickening1

nofZ .

2. More generally, if a group! acts properly discontinuously and with compact quotient by

isometries on a complete locally compact length metric space (X,dist ) then ! is ﬁnitely!

generated [BrH, Theorem I.8.10, p. 135] and ! endowed with any word metric is quasi-

isometric to (X,dist ).!

Consequently two groups acting as above on the same length metric space are quasi-

isometric.

3. Givenaﬁnitelygenerated groupGandaﬁniteindexsubgroupG init,GandG endowed1 1

with arbitrary word metrics are quasi-isometric.

In terms of Riemannian manifolds, if M is a ﬁnite covering of N then ! (M) and ! (N)1 1

are quasi-isometric.

4. Given a ﬁnite normal subgroup N in a ﬁnitely generated group G, G and G/N (both

endowed with arbitrary word metrics) are quasi-isometric.

Thus, in arguments where we study behaviour of groups with respect to quasi-isometry,

we can always replace a group with a ﬁnite index subgroup or with a quotient by a ﬁnite

normal subgroups.

5. All non-Abelian free groups of ﬁnite rank are quasi-isometric to each other. This follows

from the fact that the Cayley graph of the free group of rank n with respect to a set of n

generators and their inverses is the regular simplicial tree of valence 2n.

Now all regular simplicial trees of valence at least 3 are quasi-isometric. We denote byTk

the regular simplicial tree of valence k and we show that T is quasi-isometric to T for3 k

every k% 4.

We deﬁne the map q : T !T as in Figure 3, sending all edges drawn in thin lines3 k

isometrically onto edges and all paths of lengthk#3 drawn in thick lines onto one vertex.

The map q thus deﬁned is surjective and it satisﬁes the inequality

1

dist(x,y)#1$ dist(q(x),q(y))$ dist(x,y).

k#2

6. Let M be a non-compact hyperbolic two-dimensional orbifold of ﬁnite area. This is the

2same thing as saying that M =!\H , where ! is a discrete subgroup of PSL (R) with2R

fundamental domain of ﬁnite area.

Nota bene: We assume that all the actions of groups by isometries on spaces are to the

left, as in the particular case when the space is the Cayley graph. This means that the

41

3 1 1

2

1 3

2 1

1 1

q

5

1

11

22

3131

TT3 6

Figure 3: All regular simplicial trees are quasi-isometric.

quotient will bealways taken to the left. We feel sorry for all people which are accustomed

to the quotients to the right.

We can apply the following result.

Lemma 1.2 (Selberg’s Lemma). A ﬁnitely generated group which is linear over a ﬁeld

of characteristic zero has a torsion free subgroup of ﬁnite index.

We recall that torsion free group means a group which does not have ﬁnite non-trivial

subgroups. For an elementary proof of Selberg’s Lemma see [Al].

We conclude that ! has a ﬁnite index subgroup ! which is torsion free. It follows that1

2N=! \H is a hyperbolicsurface which is a ﬁnite covering ofM, hence it is of ﬁnite area1 R

but non-compact. On the other hand, it is known that the fundamental group of such a

surface is a free group of ﬁnite rank (see for instance [Mass]).

Conclusion: Thefundamentalgroupsofallhyperbolictwo-dimensionalorbifoldsarequasi-

isometric to each other.

Atthispointonemaystartthinkingthatthequasi-isometryistooweakarelationforgroups,

and that it does not distinguish too well between groups with di"erent algebraic structure. It

goes without saying that we are discussing here only inﬁnite ﬁnitely generated groups, because

we need a word metric and because ﬁnite groups are all quasi-isometric to the trivial group.

We can start by asking if the result in Example 6 is true for any rank one symmetric space.

Question 1.3. Given M and N orbifolds of ﬁnite volume covered by the same rank one sym-

metric space, is it true that ! (M) and ! (N) are quasi-isometric ?1 1

It is true if N is obtained from M by means of a sequence of operations obviously leaving

the fundamental group! = ! (M) in the same quasi-isometry class :1

5• goingupordownaﬁnitecovering,whichattheleveloffundamentalgroupsmeanschanging

! with a ﬁnite index subgroup or a ﬁnite extension;

• replacing a manifold with another one isometric to it, which at the level of groups means

gchanging! with a conjugate! , where g is an isometry of the universal covering.

Above we have used the following

gNotation: For A an element or a subgroup in a group G and g"G, we denote by A its image

!1gAg under conjugacy by g.

In the sequel we also use the following

Convention: In a group G we denote its neutral element by id if we consider an action of the

group on a space, and by 1 otherwise.

2 Rigidity of non-uniform rank one lattices

It turns out that the answer to the Question 1.3 is “very much negative”, so to speak, that

is: apart from the exceptional case of two dimensional hyperbolic orbifolds, in the other cases

ﬁnite volume rank one locally symmetric spaces which are not compact have quasi-isometric

fundamental groups if and only if the locally symmetric spaces are obtained one from the other

by means of a sequence of three of the operations described previously. More precisely, the

theorem below, formulated in terms of groups, holds.

2.1 Theorems of Richard Schwartz

We recall that a discrete group of isometries! of a symmetric spaceX such that!\X has ﬁnite

volume is called a lattice. If!\X is compact, the lattice is called uniform, otherwise it is called

non-uniform.

Theorem 2.1 (R. Schwartz, [Sch ]). (1) (quasi-isometric lattices)LetG beanon-uniform1 i

lattice of isometries of the rank one symmetric spaceX,i=1,2. Suppose thatG is quasi-i 1

isometric to G . Then X =X =X and one of the following holds:2 1 2

2(a) X =H ;R

g g(b) there exists an isometry g of X such that G *G has ﬁnite index both in G and in21 1

G .2

(2) (ﬁnitely generated groups quasi-isometric to lattices) Let # be a ﬁnitely generated group

2and let G be a non-uniform lattice of isometries of a rank one symmetric space X =H .R

If# is quasi-isometric to G then there exists a non-uniform lattice G of isometries of X1

and a ﬁnite group F such that one has the following exact sequence:

0!F !#!G ! 0.1

The notion of commensurability is recalled in Section 8. The particular case of commensu-

rability described in Theorem 2.1, (1), (b), means that the locally symmetric spacesG \X and1

G \X have isometric ﬁnite coverings.2

We note that Theorem 2.1 is in some sense a much stronger result than Mostow Theorem.

MostowTheoremrequirestheisomorphismoffundamentalgroups-whichisanalgebraicrelation

between groups, also implying their quasi-isometry. Theorem 2.1 only requires that the groups

6

+are quasi-isometric, which is a relation between “large scale geometries” of the two groups, and

has a priori nothing to do with the algebraic structure of the groups.

Since Mostow rigidity holds for all kinds of lattices, a ﬁrst natural question to ask is:

Question 2.2. Are the two statements in Theorem 2.1 also true for uniform lattices ?

Concerningstatement(1),thefollowingcanbesaid: givenG uniformlatticesofisometriesofi

therankonesymmetricspacesX,i=1,2,G quasi-isometric toG impliesthatX =X =X.i 1 2 1 2

g2Now one can ask if in case X = H there exists an isometry g of X such that G *G21R

ghas ﬁnite index both in G and in G ? In other words is it true that all uniform lattices of21

2isometries of the same rank one symmetric space X =H are commensurable ?

R

A weaker variant of the previousquestion is whether allarithmetic uniformlattices of isome-

2tries of X =H are commensurable.

R

The answer to both questions is negative, as shown by the following counter-example.

Counter-example:

All the details for the statements below can be found in [GPS].,

2 2 2Let Q be a quadratic form of the type 2x #a x #···#a x , where a are positive1 n in+1 1 n

rational numbers. The set

n+1H ={(x ,...,x )"R |Q(x ,...,x ) =1,x > 0}Q 1 n+1 1 n+1 n+1

is a model of the hyperbolic n-dimensional space. Its group of isometries is SO (Q), theId

connected component containing the identity of the stabilizer of the form Q in SL(n+1,R).,

The discrete subgroup G =SO (Q)*SL(n+1,Z( 2)) is a uniform lattice. Now if two suchQ Id , ,

lattices G and G are commensurable then there exist g"GL(n+1,Q[ 2]) and ""Q[ 2]Q Q1 2

such that Q 'g = "Q . In particular, if n is odd then the ratio between the discriminant of1 2 ,

Q and the discriminant of Q is a square inQ[ 2]. It now su$ces to take two forms such that1 2

this is not possible, for instance (like in [GPS]):

, ,

2 2 2 2 2 2 2 2Q = 2x #x #x #···#x and Q = 2x #3x #x #···#x .1 2n+1 1 2 n n+1 1 2 n

Statement (2) of Theorem 2.1, on the other hand, also holds for uniform lattices. See the

discussion in the beginning of Section 3.

A main step in the proof of Theorem 2.1 is the following rigidity result, interesting by itself.

Theorem2.3(RigidityTheorem[Sch ]). Let! andG be two non-uniform lattices of isome-1

n 2tries of H = H . An (L,C)–quasi-isometry q between ! and G is at ﬁnite distance from anF R

n g gisometry g of H with the property that ! *G has ﬁnite index in both ! and G.F

The meaning of the statement “q is at ﬁnite distance from g” is the following:

nFor every compact K in G\H there exists D =D(L,C,K,!,G) such that for every x with0F

Gx "K, one has0

dist(q(#)x , g#x )$D,-#"!.0 0

As it is, it does not look very enlightening. We shall come back to this statement in Section

2.3, after recalling what is the structure of ﬁnite volume real hyperbolic manifolds in Section

2.2. Also in Section 2.3 we shall give an outline of the proof of Theorem 2.3 in the particular

n 3case when H = H . All the ideas of the general proof are already present in this particularF R

case, and we avoid some technical di$culties that are irrelevant in a ﬁrst approach.

According to Selberg’s Lemma, we may suppose, without loss of generality, that both! and

G are without torsion.

7

++++2.2 Finite volume real hyperbolic manifolds

Let M be a ﬁnite volume real hyperbolic manifold, that is a manifold with universal covering

nH , for some n% 2. Let! be its fundamental group.R

Given a point x"M denote by r(x) the injectivity radius of M at x.

For every$> 0 the manifold can be decomposed into two parts:

• the $-thick part: M ={x"M |r(x)% $};""

• the $-thin part: M ={x"M |r(x)<$}.<"

The following theorem describes the structure of M. We refer to [Th, §4.5] for details.

Theorem 2.4. (1) There exists a universal constant $ = $ (n) > 0 such that for every0 0

ncomplete manifold M of universal covering H and of fundamental group !, the $ -thin0R

part M is a disjoint union of<"0

– tubular neighbourhoods of short closed geodesics;

– neighbourhoods of cusps, that is sets of the form ! \Hbo(%), where Hbo(%) is an#

nopen horoball of basepoint %" & H and ! is the stabilizer of % in !.# #R

(2) A complete hyperbolic manifold M has ﬁnite volume if and only if for every$> 0 the

$-thick part M is compact.""

Note that the fact that M is compact implies that""0

– M has ﬁnitely many components;<"0

– for every neighbourhood of a cusp, ! \Hbo(%), its boundary ! \H(%), where H is# # #

the boundary horosphere of Hbo(%), is compact.

Let now M be a ﬁnite volume real hyperbolic manifold and! =! (M). Consider the ﬁnite1

set of tubular neighbourhoods of cusps

{! \Hbo(% )|i"{1,2,...,m}}.# ii

According to Theorem 2.4, the set

m"

M =M \ ! \Hbo(% )0 # ii

i=1

is compact. The pre-image of each cusp ! \Hbo(% ) is the !–orbit of Hbo(% ) and the open# i ii

horoballs composing this orbit are pairwise disjoint. Thus, the space

m" "

3X =H \ #Hbo(% )0 iR

i=1$$!/!!i

satisﬁes!\X =M .0 0

Remarks 2.5. The group! endowed with a word metric dist is quasi-isometric to the spacew

X with the length metric dist according to the example (2) of quasi-isometry given in Section0 !

1.2. In particular, since for every x " X,!x is a net in X , we also have that (!, dist ) is0 0 0 0 w

quasi-isometric to (!x , dist ).0 !

8!

Hb 3\ ! M = H" \Hb ! "!

\ X" 0

X

0

Figure 4: Finite volume hyperbolic manifold, space X .0

2.3 Proof of Theorem 2.3

3According to Section 2.2, there exists X complementary set inH of countably many pairwise0 R

disjoint open horoballs such that !\X is compact. Consequently, ! with any word metric is0

quasi-isometric toX withthelength metric. Similarly, onecan associate toGacomplementary0

set Y of countably many pairwise disjoint open horoballs such thatG\Y is compact and such0 0

that G with a word metric is quasi-isometric to Y with the length metric.0

Thequasi-isometry q:!!Ginducesaquasi-isometry between (!x ,dist )and(Gy ,dist ),0 0! !

for every x "X ,y "Y , hence also between X and Y (each quasi-isometry having di"erent0 0 0 0 0 0

parameters L and C). For simplicity, we denote all these quasi-isometries by q and all their

constants by L and C.

In theseterms, theconclusion of Theorem 2.3 means thatq seen as aquasi-isometry between

X and Y is at distance at mostD from the restriction toX of an isometryg in Comm(!,G),0 0 0

where D =D(L,C,X ,Y ).0 0

We now give an outline of proof of Theorem 2.3.

Step 1. The following general statement holds.

Lemma 2.6 (Quasi-Flat Lemma [Sch ], §3.2). Let! be a non-uniform lattice of isometries1

3of H . For every L% 1 and C % 0 there exists M = M(L,C) such that every quasi-isometric

R

embedding

2q :Z !!

has its range in N (#! ), where ! is a cusp group and #"!.M # #

We shall come back to the proof of Lemma 2.6 later (when discussing Theorem 5.8). For the

moment let us apply it to the (L,C)–quasi-isometry q from ! to G, and to its quasi-converse

¯q :G!!.

3 3• For every #"! and %" & H corresponding to a cusp of!\H ,# R R

q(#! ).N (gG ),# M %

3 3for some g"G and '" & H corresponding to a cusp of G\H ;# R R

93 3• For every g"G and '" & H corresponding to a cusp of G\H ,# R R

%¯q(gG ).N (#! !),% M #

% % 3 3for some # "! and % " & H corresponding to a cusp of!\H .# R R

Combining both and noticing that if the left class #! is contained in the tubular neigh-#

%bourhood of another left class #! ! then the two coincide, a bijection is obtained, between left#

classes #! and left classes gG , such that# %

%dist (q(#! ),gG )$M . (2)H # %

Here dist denotes the Hausdor" distance (see the Dictionary for a deﬁnition).H

The situation is represented in Figure 5.

q

XX 00

2Figure 5: Quasi-isometric embeddings ofZ .

Step 2. The map q seen as a quasi-isometry between !x , net in X and Gy , net in Y ,0 0 0 0

3 3is extended to a quasi-isometry q between a net N in H and a net N in H . This is donee 1 2R R

horoball byhoroball. Let#! andgG satisfying (2). Let#Hb andgHb bethecorresponding# % # %

horoballs. We divide each of them into strips of constant width, by means of countably many

horospheres. We note that #! x is (-separated, and that the horosphere #H is contained in# 0 #

N (#! x ), for some(> 0 and)> 0.& # 0

(1)% % %We project #! x onto the ﬁrst horosphereH . We get a (( ,))–net, N , for some ( <(# 0 1 1

(1) (1) (1)%and ) <). We choose a maximal (-separated subset N in N , hence a ((,()–net in N1 1 1

(1)% !1and a ((,( +))–net in H . We extend q to N by q(n )=!'q'! (n ), where ! denotes1 1 11

the projection in each of the spaces onto the ﬁrst horosphereH .1

WerepeattheargumentandextendqtoanetinH,H,etc. Aglobalquasi-isometryisthus2 3

%obtained. Indeed, given two points A"H and B"H , with m%n, if B is the projection ofn m

% %B onto H , the distance dist(A,B) is bi-Lipschitz equivalent to dist(A,B )+dist(B,B).n

3 3We ﬁnally obtain a quasi-isometry q between nets ofH , hence a quasi-isometry ofH .e R R

Nota bene: In the case of the Mostow rigidity theorem also a quasi-isometry of the whole

space is obtained, but it has the extra property that it is equivariant with respect to a given

isomorphism between the two groups ! and G. Here, the property of equivariance is replaced

10