Universit`
a di Pisa
Dipartimento di Matematica
Quasi-Isometric Rigidity for
Universal Covers of Manifolds
with a Geometric
Decomposition
7 novembre 2018
Tesi di Dottorato
Candidato
Relatore
Prof.
Kirill Kuzmin
Roberto Frigerio
Introduction
This work is a drop in the stream of research originated by the ideas of Gromov in [Gro83], who pointed the attention on large scale properties of metric spaces and other mathematical objects that can be related to them, in particular finitely presented groups.
The most studied morphisms in this settings are quasi-isometric embed-dings, i.e. maps f between metric spaces (E1, d1) and (E2, d2) for which there
are constants K > 1 and c > 0 such that the inequality
K−1d1(p, q) − c 6 d2(f (p) , f (q)) 6 Kd1(p, q) + c
holds for every p, q in E.
A isometric embedding is a isometry if there exists a quasi-isometric embedding g : E2 → E1 such that the compositions f ◦ g and g ◦ f
are at a bounded distance from the identity maps of E2 and E1 respectively.
If a group G is generated by a finite set S, one can define d (x, y) for x, y in G to be the minimal length of a string of elements of S and inverses of elements of S that represents x−1y. This turns out to be a distance in G, invariant by left multiplication, and well defined up to bi-Lipschitz equivalence, hence quasi-isometry, if one changes the generating set. This is the starting point of Geometric Group Theory. One of the questions it tries to answer is whether certain algebraic properties of groups, or geometric properties of spaces on which these groups act nicely, are preserved under quasi-isometry.
A group is quasi-isometric to a finite index subgroup, and to any of its extensions by a finite group. These two steps generate the relation of virtual isomorphism between groups, and it follows that virtually isomorphic groups cannot be distinguished by quasi-isometries.
We can then define a class of groups to be quasi-isometrically rigid if any group quasi-isometric to a group of the class is virtually isomorphic to a possibly different group in the class. We also say that a group is quasi-isometrically rigid if any group quasi-isometric to it is virtually isomorphic to it.
In this thesis we address in particular the question of quasi-isometric rigidity for families of fundamental groups of certain manifolds having a canonical geometric decomposition. This decomposition comes either from the construction of the manifold, or from some operation performed on the manifold.
An example of the first case are cusp-decomposable manifolds defined by Nguyˆen-Phan in [Pha12]. They are constructed by gluing along affine
homeomorphisms of the boundary manifolds obtained by cutting away cusps of complete finite volume negatively curved locally symmetric manifolds. The precise definition will be given in Section 3. For this class of manifolds I present original results, the main of which are the following:
Theorem 0.0.1 (Theorem 3.4.20). Let Γ be a group quasi-isometric to the fundamental group of a cusp-decomposable manifold M . Then Γ is virtually isomorphic to the fundamental group of a cusp-decomposable orbifold. Theorem 0.0.2 (Theorem 3.4.21). The class of orbifold fundamental groups of cusp-decomposable orbifolds is quasi-isometrically rigid, i.e. every group quasi-isometric to an orbifold fundamental group of a cusp-decomposable orbifold is actually virtually isomorphic to the fundamental group of a, maybe different, cusp-decomposable orbifold.
These results are inspired by those proven for certain higher dimensional graph manifolds studied by Frigerio, Lafont and Sisto in [FLS15].
An example of decomposition coming from an operation on the manifold is the canonical geometric decomposition for closed non-positively curved manifolds along flat codimension 1 submanifolds, described by Leeb and Scott in [LS00]. It somewhat generalizes the JSJ decomposition in the 3-dimensional case.
Fundamental groups of cusp-decomposable manifolds turn out to be rel-atively hyperbolic, which greatly simplifies arguments in this case. We use here the definition of relative hyperbolicity using asymptotic cones following Drut¸u and Sapir in [DS+05]. Our exposition relies on some insight on the
Model Theory underlying the asymtptotic cone construction. This exposition choice is uncommon in the literature on the subject.
Relative hyperbolicity no longer holds for non-positively curved mani-folds. We explore here the large scale geometric structure of non-positively curved manifolds with non-trivial Leeb-Scott decomposition and state what is missing to prove a quasi-isometric rigidity result in higher dimensions, similar to the one of Kapovich and Leeb from [KL97] for dimension 3. More precisely, we propose some conjectures on the quasi-isometric rigidity of the Leeb-Scott decomposition, and we prove some results that could prove useful to establish our conjectures.
In Section 1 we review the theory necessary to this work. In particular, in Subsection 1.3 we describe the model-theoretic tools necessary to introduce non-standard universes, which we will use to construct asymptotic cones in the following subsection.
In Section 2 we will prove some technical results on negatively curved spaces which will turn useful in the proof of connectedness results in later sections.
In Section 3 we present the results on quasi-isometry for universal covers of cusp-decomposable manifolds. The exposition goes through four subsec-tions. The first one is introductory and describes the construction of cusp-decomposable manifolds. The second and the third ones are more geometric in nature and prove the main technical results which will be used in the fourth subsection in the proof of the main theorems.
Section 4 is an exposition on what is known and what is still unknown on non-positively curved manifolds. After a short introduction, their large scale structure is the object of the second subsection, while the third one is dedicated to some conjectures, whose plausibility derives from similar results in analogous settings.
Contents
Introduction 1
Table of contents 4
1 Preliminaries 5
1.1 Metric spaces and group actions . . . 5
1.2 Large scale geometry . . . 6
1.3 Non-standard universes . . . 9
1.3.1 Non-standard universes of metric spaces . . . 14
1.4 Asymptotic cones . . . 15
1.5 Nilpotent groups . . . 19
1.6 CAT condition . . . 22
1.7 Non-positive curvature . . . 23
2 Locally symmetric spaces 28 3 Quasi-isometric rigidity of cusp-decomposable manifolds 33 3.1 Cusp-decomposable manifolds . . . 33
3.1.1 Pieces . . . 33
3.1.2 Gluings . . . 34
3.2 The geometry of the universal cover . . . 35
3.3 Large scale structure and relative hyperbolicity . . . 37
3.4 Quasi-isometry results . . . 40
3.4.1 Walls and chambers are quasi-preserved . . . 41
3.4.2 Quasi-isometric rigidity of groups . . . 46
3.5 Open questions . . . 51
4 Quasi-isometric rigidity of non-positively curved manifolds 54 4.1 Introduction . . . 54
4.2 Large scale geometry . . . 56
4.3 Conjectured asymptotic structure . . . 63
1
Preliminaries
We recall some useful tools and set the notation.
1.1
Metric spaces and group actions
Throughout this text, the letter e will usually denote the identity of a group G. If g is an element of a group, we will denote by g also the left multiplication by it, which may have additional structure, e.g. of a diffeomorphism for a Lie group. If G acts on a set, the action of g ∈ G will be denoted by g·.
The letter d will always denote the distance function in a metric space. By this letter we will also denote the distance d (a, A) of a point a from a subset A in a metric space, and also the distance d (A1, A2), the infimum
of distances between points p1 ∈ A1 and p2 ∈ A2, not to be confused with
the Gromov-Hausdorff distance dGH between A1 and A2. The closed
R-neighbourhood of a subset A, for some positive real number R, is the set of all points at a distance at most R from A. Open neighbourhoods are defined analogously with the use of the strict inequality.
We will use notations B (x, R) and B (x, R) respectively for open and closed balls of center x and radius R in a metric space.
We now define classes of metric spaces of our interest.
Definition 1.1.1. A metric space is proper if any bounded closed subset is compact.
Recall that a geodesic is an isometric embedding of an interval of the real line into a metric space. We will use the terms geodesic segment, geodesic ray and geodesic line for geodesics defined respectively on bounded closed intervals, on a half-line or on the whole R. With an abuse of notation, we will often confuse for brevity a geodesic with its image.
Definition 1.1.2. A metric space is called geodesic if for any pair p and q of its points there is a geodesic that joins p and q. It is uniquely geodesic if such a geodesic is unique.
We will often denote a geodesic segment between p and q with pq. By the Hopf-Rinow theorem, complete Riemannian manifolds are geodesic. Definition 1.1.3. A metric space is called locally geodesic if any point has a neighbourhood which is geodesic when endowed with the restriction of the distance function.
Definition 1.1.4. A subset of a geodesic metric space is convex if any geodesic between any two of its points is contained in the subset.
Definition 1.1.5. A real valued function on a geodesic metric space is con-vex if the composition with any geodesic is a concon-vex function.
We also define angles.
Definition 1.1.6. Let (E, d) be a geodesic metric space and let γ1: [0, ε) →
E, γ2: [0, ε) → E be geodesics with γ1(0) = γ2(0). The angle between γ1
and γ2, if it exists, is lim sup (t,t0)→(0,0) arccos d (γ1(t) , γ2(t 0))2− t2− t02 2tt0 !
The following coherence result holds; (see [BH99, II, Corollary 1A.7]). Lemma 1.1.7. The Riemannian angle between two geodesics in a Rieman-nian manifold is equal to the angle defined above, called Aleksandrov angle.
Let us also introduce a couple words of terminology of group actions. Definition 1.1.8. An action of a group G on a locally compact topological space X is said to be properly discontinuous if for every compact subset K of X there is only a finite number of elements g of G such that g · K ∩ K is non-empty.
Definition 1.1.9. An action of a group on a topological space is said to be cocompact if there is a compact subset whose translates via the group action cover the whole space.
Definition 1.1.10. An action of a group G on a metric space is said to be geometric if it is by isometries, properly discontinuous and cocompact.
1.2
Large scale geometry
The main idea behind large scale geometry is a notion of equivalence between metric spaces weaker than isometry, which we are now going to define. Definition 1.2.1. Let E, F be metric spaces. A function f : E → F is a (K, c)-quasi-isometric embedding for some costants K > 1 and c > 0 if the following inequality
K−1d (p, q) − c 6 d (f (p) , f (q)) 6 Kd (p, q) + c holds for every p, q in E.
A quasi-isometric embedding is said to be a quasi-isometry if one of the following two equivalent conditions holds:
• Existence of a quasi-inverse: there exists a quasi-isometric embedding g : F → E such that the distances d (x, g ◦ f (x)) and d (y, f ◦ g (y)) are uniformly bounded;
• Quasi-surjectivity: There exists a constant D such that any point in F is at a distance at most D from some point in f (E).
It is worthwile to note that, despite the name, a quasi-isometric embed-ding does not need to be injective, neither continuous. In fact, only a sort of large scale injectivity holds: points that are far in E have images that are far in F .
Before giving any concrete example of quasi-isometry, we introduce an-other definition, which shows that large scale geometry is not a matter of metric spaces alone.
Definition 1.2.2. Let Γ be a group generated by a finite set S. We will assume for simplicity that x belongs to S if and only if x−1 does. The Cayley graph C (Γ, S) of Γ with respect to the generating set S is a graph vith vertex set Γ and an edge between g and h if and only if g−1h is in S.
A distance function on the Cayley graph is obtained by setting the length of any edge equal to 1 and by taking the induced path distance; the restriction of this distance to Γ is called word metric; in fact d (g, h) is the minimum length of a word of elements of S representing g−1h.
The precise distance obviously depends on the generating set, but not from the quasi-isometric point of view. If Γ is a group and S1 and S2 are
two different finite sets of generators, then the Cayley graphs C (Γ, S1) and
C (Γ, S2) are quasi-isometric. In particular, the identity of Γ is a bi-Lipschitz
equivalence between the two induced word metrics.
Quasi-isometries do not distinguish a finitely generated group from a finitely generated subgroup of finite index. Also they do not distinguish a finitely generated group from any of its quotients by finite normal subgroups. We sum up this in the following
Definition 1.2.3. Two finitely generated groups G and H are virtually iso-morphic if there is a finite sequence G = G0, G1, . . . , Gm = H of finitely
generated groups such that for any i = 1, . . . , m the groups Gi−1 and Gi
are obtained one from the other either by taking a finite index subgroup or by taking an extension by a finite group.
This means that virtually isomorphic groups cannot be distinguished by quasi-isometries.
A useful example of quasi-isometry, probably the most useful, is the fol-lowing fact; see [DK15, Theorem 5.29].
Lemma 1.2.4 (Milnor-ˇSvarc Lemma). Let (E, d) be a proper geodesic metric space and G a group geometrically acting on it. Then G is finitely generated by a set S and for any x0 in E the function f sending an element g of G to
g · x0 is a quasi-isometry between G and (E, d).
Groups can act on metric spaces by quasi-isometries.
Definition 1.2.5. Fix non-negative real constants K > 1, c and D. A (K, c, D)-quasi-action by quasi-isometries of a group Γ on a metric space E is a map ρ from Γ to the set of quasi-isometries of E such that:
• Every quasi-isometry in the image of Γ is a (K, c)-quasi-isometry with D-dense image;
• If e is the identity element of H, the distances d (x, ρ (e) (x)) for x in E are bounded by D;
• For any g, h in Γ, the distances d (ρ (gh) (x) , ρ (g) (ρ (h) (x))) for x in E are bounded by D.
The quasi-action is said to be cobounded if, equivalently, the orbit of some point is D0-dense for some D0 or the orbit of any point is D00-dense for some D00.
An easy example of a cobounded quasi-action is the following; see [DK15, Lemma 5.60].
Lemma 1.2.6. Suppose Γ is a finitely generated group, E is a metric space, f : Γ → E is a quasi-isometry and ϕ : E → Γ is a quasi-inverse. Then the map that sends g to the quasi-isometry f (gϕ (·)) is a cobounded quasi-isometric quasi-action.
There is also a version of the Milnor-ˇSvarc Lemma for quasi-actions; see [FLS15, Lemma 1.4].
Lemma 1.2.7. Let E be a geodesic metric space and x0 a point in E. Let G
be a group which has a cobounded quasi-action on E. Suppose furthermore that for any positive r the set {γ ∈ G|γ · B (x0, r) ∩ B (x0, r) 6= ∅} is finite.
Then G is finitely generated and the map between G and E that sends γ in γ · x0 is a quasi-isometry.
1.3
Non-standard universes
The study of the large scale geometry of metric spaces takes great advantage of model theoretic tools, in particular non-standard universes. We first recall some basic definitions, and then construct non-standard metric spaces and, based on them, asymptotic cones.
Definition 1.3.1. A language, which we will denote by L or similar symbols, is a collection of symbols of three types: functions, relations and constants.
Each function and relation symbol has associated a positive integer num-ber denoting the numnum-ber of its arguments. If k is the numnum-ber associated to a function symbol, respectively a relation symbol, we will say it is a k-ary function symbol, respectively relation symbol.
As a convention of model theory, a language is said to be countable if it is at most countable.
Languages are interpreted in models.
Definition 1.3.2. A model M for a language L is given by a set A, called universe, and an interpretation I. The interpretation associates to any k-ary function symbol F a function f : Ak → A, to any k-ary relation symbol a subset of Ak and to any constant symbol an element of A.
Example. A language for ordered fields has binary function symbols “+” and “·”, unary function symbols “−” and “−1”, a binary relation symbol “6” and constant symbols “0” and “1”. Real numbers are a model for this language with the usual interpretation.
We are now going to define formulas in a given language L. Formulas will be particular strings of symbols in the language, parentheses, variables, log-ical connectives “and” ∧ and “not” ¬, quantifier “for all” ∀ and the identity symbol “=”. We first construct terms.
Definition 1.3.3. A string of constant and function symbols of L, variables and parentheses is a term if and only if it can be constructed in a finite number of steps each involving one of the following:
1. A single variable or constant symbol is a term;
2. If F is a k-ary function symbol and t1, . . . , tk are terms, then
F (t1, . . . , tk) is a term;
We can then construct formulas using terms and the other symbols listed above.
Definition 1.3.4. A string of symbols as above is a formula if and only if it can be constructed in a finite number of steps each involving one of the following:
1. If t and s are terms, then t = s is a formula;
2. If P is a k-ary relation symbol and t1, . . . , tk are terms, then
P (t1, . . . , tk) is a formula;
3. If p is a formula, then (¬p) is a formula.
4. If p and q are formulas, then (p ∧ q) is a formula.
5. If x is a variable and p is a formula, then (∀x) p is a formula.
Parentheses will often be omitted in the sequel if no ambiguity arises. Note also that other standard logical symbols, such that “or” ∨ and “exists” ∃ can be written as above; we will often use the standard symbols as an abbreviation for longer strings.
Definition 1.3.5. A variable in a formula which is in the domain of a quan-tifier is said to be bounded. A variable which is not bounded is said to be free.
Definition 1.3.6. A formula without free variables is called sentence. If a formula p has free variables among v1, . . . , vk we will often denote it
as p (v1, . . . , vk), even if not all of these variables are actually used in p.
Given a model M there is a standard way to decide whether a given sentence σ is true in the model M. If this is the case, we say, equivalently, that the sentence σ holds, or is satisfied, in M, or that the model M satisfies σ or it is a model for σ. In symbols, M |= σ.
Definition 1.3.7. Two models M and M0 of the same language are said to be elementarily equivalent if and only if every sentence that holds in M also holds in M0, and viceversa.
Definition 1.3.8. Let p (v1, . . . , vk) be a formula with free variables among
v1, . . . , vk and M a model. A k-tuple (a1, . . . , ak) of elements of M satisfies
p, or equivalently p is satisfied by (a1, . . . , ak) if the sentence p (a1, . . . , ak)
obtained by substituting every ai to the corresponding variable vi is true in
M. In that case p is satisfiable in M.
If Σ = {pj} is a family of formulas such that the free variables of each
sentence pj(a1, . . . , ak) as above is true in M. In that case Σ is satisfiable
in M.
A family Σ as above is finitely satisfiable if every finite subfamily is sat-isfiable. Equivalently, the conjunction of every finite subset of elements of Σ is satisfiable.
Definition 1.3.9. Let M and M0 be models for a given language L, with universes A and A0 respectively. A map ι : A → A0 is said to be an elementary embedding between M and M0 if
• It is injective;
• If an element a of A is the interpretation of a constant symbol c in M, then ι (a) is the interpretation of c in M0;
• If f and f0 are interpretations of the same k-ary function symbol of L
in M and M0 respectively, then for any k-tuple (a1, . . . , ak) of elements
of A the equality f0(ι (a1) , . . . , ι (ak)) = ι (f (a1, . . . , ak)) holds;
• If R and R0 are interpretations of the same k-ary relation symbol of L
in M and M0 respectively, then a k-tuple (a1, . . . , ak) of elements of A
is in R if and only if the k-tuple (ι (a1) , . . . , ι (ak)) is in R0;
• If p (v1, . . . , vk) is a formula in L and (a1, . . . , ak) is a k-tuple of
ele-ments of A, then the k-ple (a1, . . . , ak) satisfies p in M if and only if
the k-ple (ι (a1) , . . . , ι (ak)) satisfies p in M0.
The formulas we have constructed so far do not allow to talk about subsets of a universe, families of subsets and other higher order constructs. One of the standard ways to deal with this problem is to use superstructures. We need to add to the language L we are using the binary membership relation symbol “∈”. The formulas in this case will be constructed with bounded quantifiers, defined as follows, instead of standard quantifiers.
Definition 1.3.10. Let p be a formula. The bounded quantifiers (∃ x ∈ y) p and (∀ x ∈ y) p are abbreviations for ∃ x ((x ∈ y) ∧ p) and ∀ x ((x ∈ y) → p) respectively.
Consider now a set A.
Definition 1.3.11. The superstructure V (A) of A is defined recursively as follows:
• For every positive integer m the set Vm(A) is obtained as the union of
Vm−1(A) and of its powerset;
• The superstructure V (A) is
∞
[
i=0
Vi(A).
Elements of A are called individuals and elements of V (A) \ A are called sets, with respect to A if the context requires clarification.
To avoid technical complications we will always suppose there are no membership relations involving elements of A. We will say that A is a base set if it is nonempty and every a in A is disjoint from V (A). From now on, we will always construct superstructures on base sets.
Note that, by construction, whenever i < j are non-negative integers, Vi(A) is both an element and a subset of Vj(A). Furthermore, for every
non-negative integer i the set Vi(A) is both an element and a subset of
V (A).
The superstructure V (A) can now be used as a model for L with the membership symbol “∈”, which is always interpreted as the standard mem-bership.
We now introduce ultrapowers, which provide a tool to construct new models for a language from a given one. We will construct directly the superstructure version of ultrapowers, called bounded ultrapowers. We will need some preliminary definitions first.
Definition 1.3.12. A family F of subsets of an index set I is called a filter if
• The set I belongs to F and the empty set does not; • F is closed under supersets and finite intersections.
A filter U is called an ultrafilter if for every pair A, B of subsets of I such that A ∪ B belongs to U , at least one of them belongs to U too.
An ultrafilter is said to be non-principal if it contains all the subsets of I with finite complement.
An ultrafilter U is said to be countably incomplete if there is a partition of I in a countable family of subsets none of which is in U .
It follows from the definition that any non-principal ultrafilter on a count-able index set is countably incomplete.
Convention: From now on, all the ultrafilters used will be non-principal and countably incomplete, if not explicitly stated otherwise.
Definition 1.3.13. Let U be an ultrafilter on a set I, let A be a set and (ai)i∈I a sequence of elements of A indexed by I. We say that a property
of elements of A holds U -almost everywhere on (ai), shortly U -a.e., if the
subset of indices j such that the property is true for aj is in U .
For sake of brevity, sequences indexed by N will be often denoted by (ai)
instead of (ai)i∈N.
Given a superstructure, we want to define its nonstandard universe, a new construction which enjoys additional model theoretic properties. We will do this using bounded ultrapowers. We will need an infinite set I and a non-principal countably incomplete ultrafilter U on I. We first define standard ultrapowers.
Definition 1.3.14. Let A be a set. Its ultrapower ∗A with respect to the ultrafilter U is the quotient of the set of all sequences of elements of A indexed on I by the equivalence relation of U -a.e. equality.
Definition 1.3.15. Let A be a base set and let V (A) be its superstructure. The bounded ultrapower V (∗A) is constructed recursively as follows:
• The base V0(∗A) is ∗A;
• For every positive integer m, the set Vm(∗A) is the quotient of the set
of all sequences of elements of Vm(A) by the equivalence relation of
U -a.e. equality;
• The non-standard universe V (∗A) is ∞
[
i=0
Vi(∗A).
There is a standard embedding of V (A) into V (∗A) given by the classes of constant sequences. If B ∈ V (A), then the image of B via this embedding is denoted by ∗B; the star is usually omitted for brevity for the elements of A = V0(A).
Note that we can assume either that all the elements of a sequence defining an element of Vm(∗A) are in Vm−1(A) or that all of its elements are subsets
of Vm−1(A). In the first case the class of the sequence stays in Vm−1(∗A),
which we have already defined. In the second case, the class of the sequence (Bi)i∈I is a particular subset of Vm−1(∗A) made up of classes of sequences
(Xi)i∈I of elements of Vm−1(A) such that Xi ∈ Bi U -a.e. Subsets that can
be obtained this way are called internal subsets of Vm−1(∗A).
Bounded ultrapowers are an example of non-standard universes for su-perstructures. Instead of giving the complete definition of a non-standard universe we will state the results on bounded ultrapowers which are of more interest for us and consitute part of the actual definition.
Theorem 1.3.16 (Transfer principle). The embedding of V (A) into V (∗A) is a bounded elementary embedding, i.e. an elementary embedding with respect to bounded quantifiers formulas.
The proof is by induction on the structure of the formulas; see [CK90, Theorem 4.4.5].
We give some examples of this construction in basic situations.
Example. If R is a k-ary relation on A, then ∗R is a k-ary relation on ∗A.
Classes [a1,i]i∈I, . . . , [ak,i]i∈I of sequences of elements of A are in relation via ∗R if and only if R (a
1,i, . . . , ak,i) holds for U -a.e. i in I.
Example. If B, C are sets in V (A) and f : B → C is a function, then there is a function ∗f : ∗B →∗C that on the class of a sequence (b1,i)i∈I of elements
of B has value the class of the sequence (f (a1,i))i∈I.
An important property of ultrapowers is the ω1-saturation.
Theorem 1.3.17 (ω1-Saturation). Let L be a countable language. Consider
a collection of formulas Σ = {pi(v1, . . . , vk)}i∈N whose free variables are
among v1, . . . , vk. Let M be a model and ∗M an ultrapower constructed
using a countably incomplete ultrafilter. Suppore that Σ is finitely satifiable in M. Then Σ is satisfiable in ∗M.
The statement of the theorem extends to bounded ultrapowers.
From the fact that the language is countable it follows that Σ is at most countable. The proof then explicitly constructs the satisfying element in ∗M from the elements satisfying the finite conjunctions of formulas. See [CK90, Theorem 6.1.1] for a complete proof.
1.3.1 Non-standard universes of metric spaces
Here we give some examples of non-standard universes that will turn most useful in the sequel.
Definition 1.3.18. A non-standard universe of real numbers is called hy-perreal numbers.
Fix a natural number m and consider the formula 1 + · · · + 1 6 x in the ordered field language, where the sum has m terms and x is a variable. Any finite number of these formulas, with different values of m, can be con-temporarily satisfied in the language, that is, for every finite set of natural numbers there is a real number greater or equal than them all. From satura-tion it follows that hyperreal numbers contain elements that satisfy all these formulas, i.e. that are greater than any natural number. These numbers are
called infinite, their opposites are negative infinite numbers, and the recip-rocals of infinite numbers are non-zero infinitesimal numbers; they are less in absolute value than n1, for every natural n.
For every finite, i.e. non infinite, hyperreal number x there is a unique real number, called the standard part of x, and denoted by std (x) such that x − std (x) is infinitesimal.
To define metric spaces, we first need to define real numbers, thus a language for metric spaces has all the symbols of a language for real numbers. We the need to distinguish real numbers from points in the metric space. In order to do this, we add to the language two constant symbols interpreted as “the set of real numbers” and “the metric space”. Finally, we add a binary function symbol d which will be interpreted as the distance function.
To construct the model of a metric space E, we need a superstructure with base space the disjoint union of E and the real numbers R. The distance function d : E × E → R can then be defined in the superstructure.
By taking a nonstandard universe of this superstructure we get a hyper-real distance function ∗d : ∗E ×∗E → ∗R. With an abuse of notation, we will denote by ∗E the non-standard universe of a metric space, in the place of the more correct but longer V (∗(E t R)).
When we study maps f between metric spaces E and F we consider as the base set of the superstructure the disjoint union of E, F and R, and add to the language three constant symbols: one for the real numbers and two for the metric spaces. Then f is in the superstructure, and it makes sense to study ∗f : ∗E →∗F .
Convention: Many arguments in this work will rely on model theoretic results on formulas in a language for the superstructure of a metric space. In these cases, when it is clear from the context, we will simply talk about formulas.
1.4
Asymptotic cones
We define the asymptotic cone of a metric space using nonstandard universes. Definition 1.4.1. Let (E, d) be a metric space. Construct a non-standard universe ∗E. Choose x0 in ∗E and an infinite hyperreal number Λ. The
asymptotic cone C (∗E, x0, Λ), or simply CE if the construction data are
clear or inessential, is the quotient of the set of points x in ∗E such that Λ−1∗d (x0, x) is finite by the relation that identifies x and y if Λ−1∗d (x, y)
is infinitesimal. The set CE is endowed with a metric space structure, the distance between [x] and [y] being given by std (Λ−1∗d (x, y)).
The standard definition in literature is slightly different. However, it is a rewording of the one we have just given if the non-standard universe is constructed using a non-principal ultrafilter on N, as we shall show now. We first need a definition.
Definition 1.4.2. Let (pi)i∈N be a sequence in a topological space X, let L
be a point in X and U an ultrafilter on N. We say that U − lim pi = L if
for every open neighbourhood U of L the set of indices i such that pi is in U
belongs to U .
Lemma 1.4.3. The ultralimit of a sequence always exists in a compact space. It is always unique in a Hausdorff space.
For a proof, see [DK15, Lemma 7.23]
We now state the classic definition of asymptotic cone.
Definition 1.4.4. Let E be a metric space. If λ is a real number, we denote by λE the metric space with the same support as E and with all distances multiplied by λ. Let U be a non principal ultrafilter on N, let (λi)i∈N be a
sequence of positive real numbers which is not U -a.e. bounded and (xi)i∈N
be a sequence of points in E. The asymptotic cone C (E, (xi) , (λi) , U ) is
the set of all sequences (yi) of points in E such that λ−1i d (xi, yi) is U -a.e.
bounded, quotiented by the relation that identifies two sequences (yi) and (zi)
if U − lim λ−1i d (yi, zi) = 0. The distance function is given by d ([yi] , [zi]) =
U − lim λ−1i d ((yi) , (zi)).
Note that the distance function is well defined by Lemma 1.4.3.
Lemma 1.4.5. Let E, U , (λi), (xi) be as in Definition 1.4.4. Construct
the non-standard universe ∗E of E using the ultrafilter U . Let x be the class of (xi) in ∗E and λ be the class of (λi) in ∗R. Then the asymptotic cone C ∗E, x, λ as in Definition 1.4.1 is isometric to the asymptotic cone C (E, (xi) , (λi) , U ) as in Definition 1.4.4.
Proof. Note that if (ti)i∈N is a sequence of real numbers, then its U − lim
is a real number L if and only if the standard part of the element of ∗R represented by the class of (ti) is L. The proof now easily follows from the
construction of a non-standard universe with an ultrafilter.
In order to study the structure of a metric space via asymptotic cones it might be useful to see how the decomposition of the space in certain subsets induces a decomposition of its asymptotic cones in related subsets. We begin with the following
Definition 1.4.6. Let A be any subset of a non-standard universe ∗E of a metric space E. The projection of A to an asymptotic cone C (∗E, x0, Λ) is
the set of classes of points of A in this cone.
Note that the projection of A to an asymptotic cone of E might be empty even when A is not; this happens precisely when the distances of points in A from x0 are infinite when divided by Λ.
Definition 1.4.7. Let E be a metric space and let CE be an asymptotic cone of E. Let A be a family of subsets of E. The asymptotic family CA of A is the family of projections of subsets in ∗A to CE.
We state the following simple
Lemma 1.4.8. Let E be a metric space, A a family of subsets of E and CE an asymptotic cone of E. If every point of E is at a bounded distance from the union of A then CE is the union of CA.
Proof. By the Transfer principle, every point in ∗E is at a bounded distance from the union of ∗A, and the bound is the same finite number as in E. By dividing these finite hyperreal distances by an infinite Λ we get the thesis.
Asymptotic cones turn out to be useful in the study of quasi-isometries thanks to the following result.
Lemma 1.4.9. Let E1 and E2 be metric spaces. Let K > 1 and c > 0
be real numbers. Construct a non-standard universe ∗(R t E1t E2). Fix a
point x0 in ∗E1, an infinite hyperreal number Λ and let f : ∗E1 →∗E2 be a
(K, c)-quasi-isometric embedding. Then Cf : C (∗E
1, x0, Λ) → C (∗E2, f (x0) , Λ)
that sends the class of a point x to the class of the point f (x) is well defined and is a K-bi-Lipschitz embedding.
Proof. The proof follows from the definition of asymptotic cones and cal-culations. The finite error in the definition of a quasi-isometric embedding disappears because the standard part of the finite real number c divided by the infinite Λ is 0.
We now introduce the class of tree-graded geodesic spaces, that often arises when studying asymptotic cones. We first define geodesic triangles.
Definition 1.4.10. A geodesic triangle of vertices p, q, r in a geodesic metric space is the union of a geodesic between p and q, a geodesic between q and r and a geodesic between r and p. These three geodesics are called sides of the triangle.
A geodesic triangle is said to be simple if two distinct sides intersect only at endpoints.
We can now recall the following definition from [DS+05, Definition 1.10]. Definition 1.4.11. A geodesic metric space is tree-graded with respect to a family of closed geodesic subspaces if:
(1) The space is the union of the subspaces of the family;
(2) The intersection of two subspaces in the family contains at most one point;
(3) Every simple geodesic triangle is contained in one subspace of the family. We will often use the following lemma; see [DS+05, Lemma 2.13]. We
recall that a cut-point in a connected topological space is a point whose complement is disconnected.
Lemma 1.4.12. Let E be a geodesic metric space tree-graded with respect to the family A of subspaces. Suppose that subspaces in A do not have global cut-points. Then the image of any topological embedding of a path-connected space without cut-points is contained in one subspace of the family.
Now let us see how these concepts extend to asymptotic cones.
Lemma 1.4.13. If E is a geodesic metric space, then every non-standard universe∗E is geodesic, with geodesics defined on intervals of hyperreal num-bers.
Proof. The existence of a geodesic between any pair of points can be ex-pressed through a first order formula. The proof follows directly from the Transfer principle.
Corollary 1.4.14. If E is a geodesic metric space, then every asymptotic cone of E is geodesic.
Lemma 1.4.15. If (E, d) is a uniquely geodesic metric space, then every non-standard universe ∗E is uniquely geodesic.
Proof. By the transfer principle for any pair of points p ande q ofe ∗E there is a unique internal geodesic eγ between them. We must show that there are no other geodesics between p ande q. It holds in E that for any pair of points pe and q and for any third point r the equality d (p, r) + d (r, q) = d (p, q) holds if and only if r lies on the unique geodesic between p and q. This fact can be written as a first order formula. Suppose now by contradiction that another non-internal geodesic between ep and q exists. It must pass through a pointe e
r not in the image of eγ. The equality ∗d (p,er) +e ∗d (r,e eq) = ∗d (p,eq) holdse because er is on a geodesic; by the Transfer principle it is in the image of eγ, contradiction.
We now describe an important class of geodesic metric spaces (see [DS+05, Definition 3.19]).
Definition 1.4.16. A geodesic metric space E is asymptotically tree-graded with respect to a family A of subsets if any of its asymptotic cone CE is tree-graded with respect to the asymptotic family CA defined in 1.4.7.
When applied to groups, the previous definition becomes one of the many equivalent definitions of a relatively hyperbolic group [DS+05, Theorem 8.5]. Definition 1.4.17. A finitely generated group G is hyperbolic relatively to a family of finitely generated subgroups {H1, . . . , Hm} if the Cayley graph of
G is asymptotically tree-graded with respect to the family of left cosets of the Hi’s.
1.5
Nilpotent groups
Here we recall some facts on nilpotent groups. When talking about a Lie group, we will endow it with a left-invariant Riemannian metric; we will also introduce another useful metric on nilpotent Lie groups.
Definition 1.5.1. A closed manifold is said to be infranil if it is the quotient of a simply connected nilpotent Lie group by any group of isometries acting freely and properly discontinuously.
Definition 1.5.2. A group acting by isometries properly discontinuously and cocompactly, but not necessarily freely, on a simply connected nilpotent Lie group G, is said to be almost-crystallographic.
The quotient of G by the action of an almost-crystallographic group is, in general, an orbifold. It follows by Milnor-ˇSvarc Lemma that an almost-crystallographic group is quasi-isometric to G.
Definition 1.5.3. A map between simply connected nilpotent Lie groups is said to be affine if it is the composition of a Lie groups homomorphism with a left multiplication by an element of the target group. A map between orbifolds, in particular manifolds, whose universal covers are simply connected nilpotent Lie groups is affine if any lift of it to universal covers is affine.
We will need the following result; see [Dek16, Theorem 3.7].
Lemma 1.5.4. Let N1 and N2 be quotients of simply connected
nilpo-tent Lie groups G1 and G2 respectively by an isometric action of
almost-crystallographic groups Γ1 and Γ2 respectively. Suppose that there is an
isomorphism ϕ : Γ1 → Γ2. Then ϕ is induced by an affine
homeomor-phism between N1 and N2. More precisely, there is an affine isomorphism
ϕ : G1 → G2 such that for any x in G2 and for any g in Γ1 the equality
ϕ (g) · x = ϕ (g · ϕ−1(x)) holds.
We now define a class of nilpotent Lie groups that will be used in this work.
Definition 1.5.5. Let F be one of the following: • The complex numbers C;
• The quaternions H; • The octonions O.
Denote by k the dimension of F as a real vector space, and let Im F be its imaginary part, which is a (k − 1)-dimensional real vector space. Finally, let m be a positive integer. The simply connected nilpotent Lie group G (m, F) is Fm× Im F with the following group operation:
(z1, . . . , zm, s) ∗ (w1, . . . , wm, t) = = z1 + w1, . . . , zm+ wm, s + t + m X i=1 Im ziwi ! .
It is easy to see that the inverse of (z1, . . . , zm, s) is (−z1, . . . , −zm, −s).
The commutator subgroup and the center of G (m, F) are both equal to Im F, and thus the group is step 2 nilpotent.
The Carnot-Caratheodory metric is a way to put a distance on a manifold in the subriemannian setting. Here we recall the definition specifically for the Lie groups defined above.
Definition 1.5.6. Let e ∈ G = G (m, F) be the identity element. Fix a positive definite scalar product b (·, ·) on V = TeFm ⊆ TeG. Consider the
distribution of subspaces on G obtained via left translations of (V, b). With an abuse of notation we will still call V the distribution and b the scalar product on it. Then for any pair of points g1 and g2 of G there is a piecewise smooth
curve joining them everywhere tangent to the distribution. The function
dCC(g1, g2) = inf Z p b (γ0(t) , γ0(t))dt ,
where the infimum is taken over all piecewise smooth curves γ with endpoints g1 and g2 and everywhere tangent to V , is a Carnot-Caratheodory distance
function on G, invariant by the action of G itself via left translations. Introducing this new distance has, however, no effect from the topological and the quasi-isometry point of view; see [Sch95, §2.6].
Lemma 1.5.7. The dCCdistance induces the usual topology G. The identity
of a nilpotent Lie group as above is a quasi-isometry between its Riemannian and Carnot-Caratheodory distances.
It is known from [Gro81] that finitely generated nilpotent groups have locally compact asymptotic cones. We now want to study asymptotic cones of almost crystallographic groups, which by Milnor-ˇSvarc Lemma are the same as asymptotic cones of nilpotent Lie groups they are acting on. The work of Pansu [Pan83] gives an explicit description for any asymptotic cone of a simply connected nilpotent Lie group G; it is a suitable nilpotent Lie group G∞ constructed from G and endowed with a Carnot-Caratheodory metric.
In our case, this description reduces to the following result; see [Pan83, (B), §49].
Lemma 1.5.8. Asymptotic cones of a nilpotent Lie group G (m, F) are all isometric to the group itself with a left-invariant Carnot-Caratheodory met-ric.
We finish this subsection with a result allowing us to distinguish groups in the class above.
Lemma 1.5.9. Consider two nilpotent Lie groups G1 and G2, each being
either some Rm or some G (m, F) and endowed with a left-invariant
Carnot-Caratheodory metric. Suppose that there is an open subset of G1 bi-Lipschitz
homeomorphic to an open subset G2. Then G1 and G2 are isomorphic as
Proof. Bi-Lipschitz homeomorphisms preserve topological and Hausdorff di-mension. These dimensions are both m for open subsets of Rm. Take now F
to be one of C, H or O and let k be the dimension of F as a real vector space. Then the topological dimension of an open subset of G (m, F) is km + k − 1 and the Hausdorff dimension is km + 2 (k − 1) by [M+85]. The lemma easily
follows.
1.6
CAT condition
We recall here a notion of curvature bounded from above for geodesic metric spaces.
Definition 1.6.1. Let n be an integer greater than or equal to 2 and κ a real number. The n-dimensional model space with curvature κ is Mnκ defined as
follows:
• The euclidean n-dimensional space if κ = 0;
• The standard n-sphere with distances divided by √κ if κ > 0; • The hyperbolic n-space with distances divided by √−κ if κ < 0.
The notion of CAT (κ) metric space involves comparisons between trian-gles in the space and triantrian-gles in model spaces.
Definition 1.6.2. If p, q and r are three points in a geodesic metric space E, a geodesic triangle 4pqr having them as vertices is the union of a geodesic between p and q, a geodesic between q and r and a geodesic between r and p. Definition 1.6.3. Let 4pqr be a geodesic triangle in a geodesic metric space E. Fix a real number κ. If κ > 0, suppose furthermore that d (p, q)+d (q, r)+ d (r, p) 6 √2π
κ. The comparison triangle is a geodesic triangle in M 2
κ, which
exists and is unique up to the action of the isometry group of M2κ, with vertices
e
p, q ande er such that d (p, q) = d (p,eq), d (q, r) = d (e eq,er) and d (r, p) = d (er,p).e Definition 1.6.4. Let E be a geodesic metric space and κ a real num-ber. Consider a geodesic triangle 4pqr in E such that, if κ > 0, then d (p, q) + d (q, r) + d (r, p) 6 √2π
κ. Consider furthermore points x ∈ pq and
y ∈ pr. On a comparison triangle 4peeqer in M2
κ consider the uniquely
deter-mined comparison points x ∈e peeq and y ∈e peer for x and y respectively such that d (p, x) = d (p,e ex) and d (p, y) = d (ep,y). The space E is CAT (κ) if fore every choice of 4pqr, x and y in it as above, the inequality d (x, y) 6 d (ex,y)e holds.
A locally geodesic metric space is locally CAT (κ) if every point has a CAT (κ) neighbourhood.
Being locally CAT (κ) for a locally geodesic metric space E is a synonim of having curvature less than or equal to κ. In particular,
Lemma 1.6.5. A Riemannian manifold is locally CAT (κ) if and only if its sectional curvature is bounded above by κ.
For a proof, see [BH99, II, Theorem 1A.6].
Corollary 1.6.6. If κ1 6 κ2 then a CAT (κ1) space is CAT (κ2) too.
The following result is analogous to the classical Cartan-Hadamard the-orem; see [BH99, II, Theorem 4.1].
Lemma 1.6.7. Let κ be a real number and let E be a complete connected locally CAT (κ) metric space. Then there exists a distance function on the universal cover eE such that the cover projection is a local isometry and eE is a CAT (κ) space.
We recall the following fact on angles in CAT (κ) spaces; see [BH99, II, Proposition 3.1].
Lemma 1.6.8. Angles always exist between geodesics in a locally CAT (κ) space.
1.7
Non-positive curvature
Here we recall some phenomena specific to the non-positive curvature. Definition 1.7.1. A geodesic metric space is said to be non-positively curved if it is locally CAT (0).
One of the most important properties of CAT (0) spaces is the convexity of the distance function; see [BH99, II, Proposition 2.2].
Lemma 1.7.2. Let E be a CAT (0) space and let γ1: [0, T1] → E and
γ2: [0, T2] → E be geodesics. Then the function defined on [0, 1] as t 7→
d (γ1(tT1) , γ2(tT2)) is convex.
Corollary 1.7.3. A CAT (0) space is uniquely geodesic.
Corollary 1.7.4. In a CAT (0) space the distance function from a closed convex subset is convex.
In CAT (0) spaces it is possible to define a closest point projection on convex subsets analogously to what happens in Hilbert spaces; see [BH99, II, Proposition 2.4].
Lemma 1.7.5. Let E be a CAT (0) space and let C be a convex subset which is complete when endowed with the restriction of the distance function. Then for every x in E there is a unique point pC(x) in C which realizes the
minimum of the distances from x among the points of C. The angle at pC(x)
between the geodesic connecting it to x and any geodesic connecting it with a point in C is at least π2. The function pC is 1-Lipschitz.
We now recall the notion of boundary at infinity for proper CAT (0) spaces.
Definition 1.7.6. Let E be a proper CAT (0) space. The boundary at in-finity of E, denoted by ∂E, is the set of geodesic rays γ : [0, +∞) → E quotiented by the relation that identifies two rays if their images lie at finite Hausdorff distance from each other. The points of ∂E are called points at infinity of E. The space E is the disjoint union of E and ∂E.
There is another equivalent definition of E, which also allows to define a topology on this space.
Definition 1.7.7. Fix a point x0 in E. Closed balls B (x0, r) centered in
x0 of radius r are convex and complete. Then a closest point projection is
defined from B (x0, r1) to B (x0, r2) whenever r1 > r2. The inverse limit of
this system of topological spaces and continuous maps does not depend on x0
and is called E.
Let us see how these two definitions are related.
Lemma 1.7.8. For every point x0in a proper CAT (0) space E and for every
ω in ∂E as in Definition 1.7.6 there is a unique geodesic ray γ : [0, +∞) → E with γ (0) = x0 in the class of ω. Then ω is embedded in E as in Definition
1.7.7 as the class of points γ (r) in B (x0, r). A point x of E is embedded in
E as in Definition 1.7.7 as x itself in balls B (x0, r) with r > d (x, x0). There
are no other points in E.
Lemma 1.7.9. If Y has curvature bounded above by some negative real number, then for any pair of points ω1 and ω2 in ∂Y there is a unique, up
to reparametrization via translations, geodesic line γ : R → Y such that the ray defined on the positive half of R is in the class of ω1 and the ray defined
on the negative half is in the class of ω2.
.
The proofs of the above lemmas on the boundary at infinity can be found in [BH99, Chapter II.8].
We recall some facts about isometries and actions of groups by isometries on non-positively curved spaces.
Definition 1.7.10. If g is an isometry of a metric space (E, d), its displace-ment function on a point x of E has the value d (x, g (x)).
Definition 1.7.11. If g is an isometry of a metric space (E, d) the subset of E where the displacement function of g attains its minimum, if non-empty, is called min (g).
In the case min (g) is non-empty and g is different from the identity, the isometry g is said to be elliptic if the minimum of the deplacement function of g is 0 and hyperbolic otherwise.
Lemma 1.7.12. If g is a hyperbolic isometry of a CAT (0) space E, the set min (g) is convex, and it is isometric to the product Y × R, where Y is a convex subset of E. The action of g on min (g) is by identity on the Y factor and by a translation on the R factor.
Definition 1.7.13. If G is a group of isometries of a CAT (0) space E, we define min (G) to be the intersection of the sets min (g) for g ∈ G.
Theorem 1.7.14. If G is a finitely generated free abelian group of rank m of hyperbolic isometries of a CAT (0) space E, the set min (G) is non-empty, convex, and it is isometric to the product Y0 × Rm, where Y0 is a convex
subset of E. The action of every g ∈ G on min (G) is by the identity on the Y factor and by a translation on the Rm factor. The set of translation
vectors corresponding to a set of free generators forms a basis for Rm.
Lemma 1.7.15. The non-identity elements of a group acting by properly discontinuously and cocompactly isometries on a proper CAT (0) space are only elliptic and hyperbolic isometries.
Lemma 1.7.16. A non-identity element of a group acting properly discon-tinuously by isometries on a proper CAT (0) is an elliptic isometry if and only if it is a torsion element.
For the proofs and other facts on group actions on CAT (0) spaces, see [BH99, Chapter II.6]
Corollary 1.7.17 (Flat Torus Theorem). Suppose that a group Γ acts prop-erly discontinuously and cocompactly by isometries on a CAT (0) space E. Suppose it has a subgroup isomorphic to Zm, then there is a locally
isomet-rically immersed flat m-torus in the quotient E/Γ.
In the smooth case the torus is totally geodesically immersed, as already proven by Gromoll and Wolf in [GW71] and Lawson and Yau in [LY72].
Lemma 1.7.18. Suppose that G is a group that acts by isometries on a CAT (0) space E and preserves a convex subset C. If min (G) is nonempty, then min (G) ∩ C is non-empty.
Proof. Let pC be the projection on C and g an element of G. It follows by
definition that for any x in E we have pC(g (x)) = g (pC(x)). The map pC
is 1-Lipschitz, so if x was taken to be a point in min (G), then also pC(x) is
in min (G).
We define and recall some facts on Busemann functions.
Definition 1.7.19. Let E be a proper CAT (0) space. Choose a geodesic ray γ in E. The Busemann function relative to γ is the real valued function b : E → R defined by b (p) = limt→+∞d (p, γ (t)) − t.
Lemma 1.7.20. Busemann functions relative to rays in the same class in the boundary at infinity differ by a constant. Therefore, it is possible to talk about Busemann functions relative to a point in the boundary at infinity. When the curvature is bounded from above by a negative real number, the limit of a Busemann function relative to ω at a point at infinity in Y is −∞ in ω and +∞ in all the other points.
Any Busemann function is convex and 1-Lipschitz. Its derivative is ex-actly −1 on geodesic rays in the class of the boundary point ω to which they are relative to. On smooth CAT (0) manifolds it has a (classical) gradient with norm equal to 1.
For proofs and other facts on Busemann functions, see [BH99, Chapter II.8]
Definition 1.7.21. A horoball is a sublevel set of a Busemann function. A horosphere is the boundary of a horoball.
We finish this section by telling how non-positive curvature behaves with respect to asymptotic cones.
Lemma 1.7.22. Any asymptotic cone CE of a CAT (0) space E is itself CAT (0).
Proof. By the Transfer principle, the CAT (0) inequality continues to hold in the non-standard universe ∗E from which CE is constructed, if we consider comparisons with ∗E2. Choose points x and y in CE, and let x0, y0 be points
of ∗E that project to x and y respectively. By convexity of the distance function in ∗E the projection of any geodesic with endpoints that project to
x and to y coincides with the projection γ of the geodesic γ between x0 and y0.
Note, however, that a priori there could be geodesics between x and y in CE which cannot arise as projections of geodesics of∗E to the cone. We wish
to show that this is not possible. This is trivial if x = y, so let the distance between the two points be a positive real number L. Consider a geodesic in CE between x and y different from the one just found. It must contain a point z not in the image of γ. Let l > 0 be the distance of z from the closed image of γ, and Λ be the infinite hyperreal number by which the distances in ∗E are divided to obtain CE. Then the distance from the closed image of γ of a point z0 of ∗E projecting to z is Λ (l + εz0), with εz0 infinitesimal. We
also have ∗d (x0, y0) = Λ (L + εx0y0) for some infinitesimal εx0y0.
Consider now a comparison triangle 4xeyez ine ∗E2 for 4x0y0z0. The dis-tance ed of ez from the side exey is realized in some pointp ofe exey; let p0 be the comparison point for p on xe 0y0. We then have that
e
d = d (ez,p) > d (ze 0, p0) > d (z0, γ) = Λ (l + εz0)
because the CAT (0) inequality holds in∗E. It is now a matter of elementary geometry, and of continuity of the square root, that
d (x0, z0) + d (z0, y0) = d (x,e z) + d (e z,e y) > Λe √L2+ 4l2+ ε
for some infinitesimal ε. By projecting the points back to the cone we get a contradiction.
The CAT (0) inequality for a triangle 4pqr in CE now easily follows from the fact it holds for any triangle in ∗E whose vertices project to p, q and r respectively.
2
Locally symmetric spaces
In this section we present a technical result on configurations of horospheres in negatively curved symmetric spaces or, more in general, on complete sim-ply connected Riemannian manifolds with curvature bounded above by some negative number. We will use strictly negatively curved symmetric spaces in this text; we recall their classification.
Lemma 2.0.1. A negatively curved Riemannian symmetric space of dimen-sion n is one of the following:
• The real hyperbolic space RHn;
• The complex hyperbolic space CHn2; this case can occur only if n is
even;
• The quaternionic hyperbolic space HHn4; this case can occur only if n
is a multiple of 4;
• The Cayley or octonionic hyperbolic plane OH2; this case can occur only if n is equal to 16.
We will concisely refer to a symmetric geometry by saying that it is n-dimensonal F-hyperbolic, with F equal to, respectively, R, C, H or O. For a proof, see [Sch95, §2.3].
The next lemma shows that a pointwise finite collection of horoballs is actually locally finite.
Lemma 2.0.2. Let Y be a simply connected complete Riemannian manifold with curvature bounded from above by some negative number. Consider in Y a collection of horoballs (Bi)i∈I such that every point of the space is contained
in at most a finite number of them. Then any closed ball intersects only a finite number of horoballs in the collection. In particular, this collection is at most countable.
In the sequel we will call for brevity “center” of a horoball the unique point on ∂Y contained in the closure in Y of the horoball itself.
Proof. A horoball is, by definition, the sublevel set of a Busemann function; call fi the Busemann function having Bi as sublevel set. Recall that the
norm of the gradient of these functions is equal to 1. For our convenience, to each fi we add some constant such that fi takes the value 0 on the boundary
Let p be the center of a ball B of positive radius R that intersects an infinite number of horoballs of the collection. Note that the fact that the norm of the gradient of a Busemann function is equal to 1 implies that if B intersects a horoball Bi of the collection then the respective Busemann
function fi has value strictly less than R in p. The sequence (ωj)j∈J, with
J ⊆ I, of the centers of horoballs intersecting B has an accumulation point ω in the compact space ∂Y .
Consider the geodesic ray α : [0, +∞) → Y starting from p and belonging to the class of ω in ∂Y , and let q be the point α (2R). The set of points r of Y such that the geodesic qr forms an angle strictly less than π3 with α|[2R,+∞) is open. By definition of topology on ∂Y the set O of classes of geodesic rays in it starting from q and forming an angle strictly less than π3 with α|[2R,+∞) is also open. Consider the set of the centers ωj0 of horoballs Bj0 of
our collection that lie in O and such that Bj0 intersects B. By construction
these ωj0 are infinite. Furthermore by the negative curvature condition in
any point q0 of α ([0, 2R]) the angle between α and a geodesic ray starting from q0 and belonging to the class of one such ωi is strictly less than π3, and
thus the derivative of fi ◦ α is strictly less than −12 on [0, 2R]. But then
fi(q) < R − 122R = 0, and thus q stays in infinitely many horoballs of the
collection, contradiction.
In what follows we will need a simple observation on the flow of a vector field that will allow us to perform continuous retractions.
Lemma 2.0.3. Let Φt be the flow of a smooth vector field on a smooth
manifold Z. Let τ be a continuous nonnegative function on Z, tought of as a “stopping time”, such that for every z in Z the flow Φt(z) is defined for
all the nonnegative times less than or equal to τ (z). Define Φ : Z → Z as Φ (z) = Φτ (z)(z) and denote by R+ the nonnegative real half line.
Then the function Ψ : Z × R+ → Z such that Ψ (z, t) = Φ
min{t,τ (z)}(z)
is a continuous retraction of Z on the image of Φ. Furthermore, if ξ is any increasing homeomorphism between the interval [0, T ] and R+∪ {+∞}, then
the function on Z × [0, T ] defined as (z, t) 7→ Ψ (z, ξ (t)) is a continuous retraction defined on the interval [0, T ].
Proof. The proof follows easily by noting that all the functions we are com-posing in our candidate retractions are continuous and by local compactness of the manifold Z.
We now state and prove the main theorem of this section. To simplify the exposition we will always think of a pair ij of natural indices as of an unordered pair.
Theorem 2.0.4. Let Y be a simply connected complete Riemannian mani-fold with curvature bounded from above by some negative number. Consider in Y a collection of horoballs (Bi)i∈N such that every triple of pairwise
dif-ferent horoballs of the collection has empty intersection. Let N0 be the com-plement in Y of the union of this collection. Suppose also that two different horoballs of the collection have distinct centers.
If the horoballs Bi and Bj intersect, let γij be the geodesic between their
centers, contained in their union. Let G be the union of the images of the γij’s for all the intersecting pairs of horoballs. Then G is a submanifold of
Y , and its complement Y0 retracts by deformation on N0.
Proof. Retain the notation fi for the Busemann function having Bi as
sub-level set, as in the proof of the previous lemma. Let furthermore ωi be the
center of Bi. Recall that in negatively curved spaces these functions are
con-vex. As before, we can suppose that fi takes the value 0 on the boundary
horosphere of Bi and negative values inside.
For the first claim, we must prove that Y0 is open in Y and that every point of a geodesic γij has a neighbourhood U such that the pair (U, G)
is homeomorphic to (Rm, R), where R is a coordinate line in Rm. Since
geodesics are isometrically embedded in Y , this is true if every point of Y has a neighbourhood intersecting only finitely many geodesics in G.
Recall that the image of γij is a subset of Bi∪ Bj. It follows that if every
neighbourhood of a point p of Y intersects infinitely many geodesics in G, by Lemma 2.0.2 every neighbourhood of p instersects infinitely many geodesics in G that share an endpoint ωi0 in ∂Y , which is the center of a horoball
Bi0 in the collection, defined as the set where the Busemann function fi0 is
negative. Recall that geodesics lines starting from ωi0 are integral lines of
the gradient flow of fi0. Consider now the geodesic β : R → Y starting from
ωi0 parametrized in such a way that fi0(α (t)) = t and passing by p. This
way the point β (0) is on the boundary of Bi0 and by the continuity of the
gradient flow of fi0 every ball of positive radius centered in it still intersects
infinitely many geodesics in G having ωi0 as endpoint at infinity. This means
that every ball of positive radius with center in β (0) intersects infinitely many horoballs of the collection, which contradicts the previous lemma.
Now we make some preliminary construction for the retraction claimed in the second part of the thesis. If two horoballs of the collection Bi and Bj
intersect, inside their intersection in Y0 there is the set Uij where fi = fj < 0.
This is a codimension 1 submanifold because it is the zero set of the function fi− fj, which has nonzero gradient. Indeed, it could have gradient equal to
0 in a point p if and only if gradpfi = gradpfj, but these two gradients are
and thus Bi should be equal to Bj. More generally, in a point of Y0 the
gradients of two different fi and fj cannot be parallel. In fact, they are
parallel only on γij where they are opposite, and we have removed those
geodesics from Y0.
Let α : (−∞, 0] → Y0 be a geodesic starting from the center of Bi
parametrized such that fi(α (t)) = t. This way the point α (0) is on the
horosphere bounding Bi. We claim that
• If α (−∞, 0) intersects one of the sets Uj1j2, then one of j1, j2 is equal
to i, say the first without loss of generality;
• The geodesic α|(−∞,0) intersects at most one of the sets Uij with j
different from i and it does so if and only if the point α (0) is inside Bj;
in this case the intersection consists of one point only.
The first item is readily proven by observing that if both j1and j2are different
from i, then the intersection of α ((−∞, 0)) with Uj1j2 lies inside Bi∩Bj1∩Bj2,
which contradicts the hypothesis. For the second, observe that the derivative of t 7→ fi(α (t)) is constantly equal to 1 and all the derivatives of t 7→
fj(α (t)) with j 6= i have absolute value strictly less than 1 because the
gradients of fi and fj are never parallel in Y0.
Suppose now that α (0) is outside Bj for a j 6= i. This means that
fj(α (0)) > 0, and by our observation on the derivatives we have that
fi(α (t)) < fj(α (t)) for any t < 0. It follows that α ((−∞, 0)) can
in-tersect Uij only if fj(α (0)) < 0, and in fact in this case it does because
limt→−∞fj(α (t)) = +∞. Also, if α (−∞, 0) intersected both Uij1 and Uij2
with j1 6= j2, then we would have that both fj1 and fj2 are strictly less than
0 in α (0), but this would force the points α (t) to stay in Bi∩ Bj1 ∩ Bj2 for
negative values of t close enough to 0, a contradiction.
Finally note that from the observation on the derivatives fi can be equal
to fj on α in at most one point, which concludes the proof of the second part
of the claim.
We can now perform the retraction. We first retract Y0 on N0∪S Uij.
If a point p lies outside N0∪S Uij, then there is an i such that fi(p) < 0
and fi(p) < fj(p) for any j 6= i. If we move p along the geodesic starting
from the center of Bi away from it, or along the gradient flow of the
Buse-mann function fi, which is the same, it reaches either N0 or some Uij; by
our previous considerations exactly one case happens. We want to define a function τi for the gradient flow of fi that respects the hypotheses of Lemma
2.0.3; then, by using this lemma, we perform the retraction along the gradi-ent of fi in the time interval [1 − 2−i, 1 − 2−i−1]. The juxtaposition of these
has a neighbourhood V0 where only a finite number of them is different from the identity, and so this juxtaposition is the identity map in V0 sufficiently near the time 1, and we can prove the continuity locally.
The retraction involving fi will be defined on Yi0, that is Y without the
geodesics of G starting from the center of Bi, and then restricted to Y0. This
is possible because the other geodesics of G stay in the subset which is left untouched by this retraction, and makes things easier because it allows us to define the geodesics starting from ωi and not in G on the whole R. Let
ϕi be the function inf {0} ∪ {fj|j 6= i}. By the Lemma 2.0.2 this is a locally
finite infimum, and thus ϕi is well defined and continuous on all of Yi0. By
construction, in the retraction along the geodesics starting from ωi we are
moving points in the subset where ϕi− fi is positive, and leaving fixed the
others. We want then to define τi as the maximum between 0 and the time
that takes to this difference to become nonpositive when moving a point along a geodesic β : R → Y starting from ωi. Note that all the functions involved
in the minimum defining ϕi, when composed with β, have derivative strictly
less than 1. This means that for every real number t0 the function ϕi ◦ β
is µ-Lipschitz in a neighbourhood of t0, with µ < 1, and thus the function
(ϕi− fi) ◦ β is strictly decreasing.
Let Φi,t be the gradient flow of fi, which is defined on all times on Yi0
by construction. The continuous function ψi: Yi0 × R → Y 0
i × R such that
ψi(y, t) = (y, (ϕi− fi) ◦ Φi,t(y)) is injective by previous considerations, and
for every y there is ty such that ψi(y, ty) = (y, 0). By the invariance of
domain ψi is locally invertible, and we define the function τi0 on Y 0 i to be
such that the inverse of (y, 0) is (y, τi0(0)). We finally set τi = max {0, τi0}.
We then retract each Uij on the set fi = fj = 0 following the gradient
flow of fi+ fj. To prove the continuity, we use a similar argument to the first
part; we juxtappose the apriori infinite numerable retractions involved and for each of them we define a stopping time. Note that the gradient of fi+ fj
is tangent to Uij because grad (fi− fj) is orthogonal to Uij by construction
and grad (fi+ fj) is orthogonal to grad (fi− fj) because grad fi and grad fj
both have norm equal to 1. We can thus define the retraction on all of Y \ γij
and then restrict it to N0 ∪S Uij. In particular, for each pair ij we will
exhibit a retraction of Y \ γij on the set where fi+ fj > 0.
On Y \ γij the gradient of fi+ fj is always nonzero; indeed on Y it is zero
only on γij, which we have removed. Furthermore, the norm of this gradient
is nondecreasing during the flow because fi+ fj is convex. This means that
starting from any point where fi+ fj is negative, the value of fi+ fj strictly
grows to 0 in a finite time. We then define, as before, the function τij(y) to
be the maximum between 0 and the time that (fi+ fj) ◦ Φ (y) takes to reach
