Orlicz spaces and the large scale geometry of Heintze groups
We consider an Orlicz space based cohomology for metric (measured) spaces with bounded geometry. We prove the quasi-isometry invariance for a general Young function. In the hyperbolic case, we prove that the degree one cohomology can be identified with an Orlicz-Besov function space on the boundary at infinity. We give some applications to the large scale geometry of homogeneous spaces with negative curvature (Heintze groups). As our main result, we prove that if the Heintze group is not of Carnot type, any self quasi-isometry fixes a distinguished point on the boundary and preserves a certain foliation on the complement of that point.
Keywords: Orlicz spaces, Heintze groups, quiasi-isometry invariants, -hyperbolicity.
2010 Subject classification: 20F67, 30Lxx, 46E30, 53C30.
In this article we are interested in the large scale geometry of Heintze groups. Homogeneous manifolds with negative sectional curvature where characterized by Heintze in [Hei74]. Each such manifold is isometric to a solvable Lie group with a left invariant metric, and the group is a semi-direct product where is a connected, simply connected, nilpotent Lie group, and is a derivation of whose eigenvalues all have positive real parts. Such a group is called a Heintze group.
A purely real Heintze group is a Heintze group as above, for which the action of on the Lie algebra of has only real eigenvalues. Every Heintze group is quasi-isometric to a purely real Heintze group, unique up to isomorphism, see [Cor12, Section 5B]. In the sequel we will focus only on purely real Heintze groups.
Let be the Lie algebra of , and be the Lie algebra of derivations on . We denote by the matrix exponential. The group structure of is then given by a contracting action , where satisfies .
We will use the notation to denote a point of . Any left invariant metric on is Gromov hyperbolic, since any two such metrics are bi-Lipschitz equivalent. Assume that is equipped with a left invariant metric for which the vertical lines are unit-speed geodesics. They are all asymptotic when , and hence, they define a “special” boundary point denoted . The boundary at infinity is a topological -sphere, and can be therefore identified with the one-point compactification . The left action of on its boundary has two orbits, namely, and .
Two general problems motivate this work: first, to understand , the group of self quasi-isometries of ; and second, the quasi-isometric classification of Heintze groups. These problems have been approached by many authors and by means of several methods, see for instance [Ham87, Pan89b, FM00, Pan07, Dym10, CT11, DP11, Pen11, Xie12, SX12, Xie14a]. We refer the reader to [Cor12] for a survey on the subject.
In this article, we focus on the Pointed Sphere Conjecture [Cor12, Conjecture 6. C. 9]. It states that is fixed by the boundary homeomorphism of any self quasi-isometry of , unless is isometric (for some left invariant metric) to a rank one symmetric space.
The conjecture is known to be true in the following cases. Recall that is said to be of Carnot type if the Lie algebra spanned by the eigenvectors corresponding to the smallest eigenvalue of is the whole algebra [Cor12, Definition 2. G. 1].
[Pan89b, Corollary 6. 9], the conjecture holds whenever is not of Carnot type and is diagonalizable.
The idea behind the proofs is similar in all the three cases. It consists in finding a quasi-isometry invariant foliation on which is singular at the point . The leaves of this foliation are the accessibility classes of points by rectifiable curves, with respect to an appropriate visual metric on the boundary. To this end, Pansu consider -cohomology, and Xie consider the -variation of functions on the boundary.
Following an original idea of Romain Tessera, and Pansu’s methods, we propose here an approach based on the theory of Orlicz spaces [RR91]. This allows us to extend these results to all Heintze groups which are not of Carnot type. We mention that Orlicz spaces based cohomologies have been considered recently also in [Kop13, KP13].
The next theorem is our main result. Let be the smallest eigenvalue of , and let be the closed connected subgroup of whose Lie algebra is spanned by the -eigenvectors belonging to the -Jordan blocks of maximal size. It is a non-trivial and proper subgroup of when is not of Carnot type, see Section 1.2 for more details.
Let be a purely real Heintze group that is not of Carnot type. The boundary homeomorphism of any self quasi-isometry of fixes the special boundary point . Furthermore, in this case, it preserves the left cosets of the subgroup .
Another important ingredient in our approach is a localization technique. Let us motivate it by considering the following examples. Let and consider the Heintze group , , where
and . Notice that is isometric to the real hyperbolic space . The degree one -cohomology of can be identified with a Besov space on the boundary [Pan89a, BP03]. Let us consider the quasi-isometry invariant Banach algebra of continuous Besov functions . The dependence on of this algebra is summarized in Figure 1.1.
One way to isolate the point is suggested by the following construction which appears in [Shc14]. Let be the Heintze cone defined as the quotient space of by the discrete group of translations . The boundary is the union of a torus and the isolated point . The dependence on of is summarized in Figure 1.2. One would like to define a similar cone with respect to any other point .
The pull-back of a function by the projection map is periodic and does not define an element of . Nevertheless, it satisfies a local integrability condition, i.e. it belongs to the Fréchet algebra . The key point is that the dependence on of the algebras coincides with that summarized in Figure 1.2 when , and with that in Figure 1.1 when . This explains the case (ii).
Let us look at (iii) in more detail. The parabolic visual metric on is bi-Lipschitz equivalent to the function
From this expression, we see that the projection functions, and , have different regularity properties with respect . That is, is Lipschitz, while satisfies the inequality
In particular, is -Hölder for any exponent . Therefore, both projections belong to for any , which explains Figure 1.2.(iii). In other words, the -cohomology in degree one is not sensible to the logarithmic term appearing in (1.1). As we will show later, a well chosen Orlicz-Besov space detects the difference between and , and we are able to distinguish in case (iii) also.
The quasi-isometry invariance of Orlicz-Besov spaces is not evident at first sight, we show it for a class of Young functions which is sufficient for our purposes in Sections 3 and 4. We deal with localization in Section 5.
As a by product, finer results regarding are obtained. Let us outline them.
1.1. On the Orlicz cohomology of a hyperbolic complex and its localization
Let be a finite dimensional simplicial complex with bounded geometry. That is, there exists a constant such that any vertex of is contained in at most simplices. Suppose is equipped with a geodesic distance making each of its simplices isometric to a standard Euclidean simplex. We further assume that is uniformly contractible: it is contractible and any ball in is contractible in the ball , for some which depends only on . In the hyperbolic case, this condition is not restrictive.
By a Young function we mean an even convex function , with and . For such a function, we introduce here the Orlicz cohomology, denoted by , of the complex . It consists on a direct generalization of the ordinary -cohomology introduced in [BP03, Gro93, Pan89a], where . As in the ordinary case, we show that it is a quasi-isometry invariant of .
Suppose that is in addition a quasi-starlike Gromov-hyperbolic space. Recall that quasi-starlike means that any point of is at uniformly bounded distance from some geodesic ray. Inspired by the works of Pansu [Pan89a, Pan02, Pan08], and Bourdon-Pajot [BP03], we identify the degree one Orlicz cohomology of with an Orlicz-Besov functional space on its boundary . To this end, we need to assume a decay condition on the Young function.
Let be a Young function. We say that is doubling, if there exist and such that .
The doubling condition in the above definition is known in the literature as the condition. It admits several equivalent formulations, see for example [RR91, Thm. 3 Ch. 2].
Moreover, suppose that the boundary at infinity admits a visual metric which is Ahlfors regular of dimension . That is, the -dimensional Hausdorff measure of a ball of radius is comparable to . We refer the reader to [BH99, GdlH90, Hei01] for basic background.
In the space of pairs , consider the measure
If is a measurable function, we define its Orlicz-Besov -norm as
Then, the Orlicz-Besov space is by definition
We denote by the functions on which are constant -almost everywhere. Then the space when equipped with the norm is a Banach space.
Let be a doubling Young function, and be an Ahlfors regular visual metric on . There exists a canonical isomorphism of Banach spaces between and . In particular, is reduced.
Notice that by [BP03, Prop. 2.1], any Ahlfors regular compact metric space is bi-Lipschitz homeomorphic to for some geometric hyperbolic complex as above, and the quasi-isometry class of depends only on the quasi-symmetry class of . In particular, Orlicz-Besov spaces are quasisymmetry invariants of . When the metric space is the Euclidean -sphere, and , this Orlicz-Besov space coincides with the classical Besov space [Tri83].
In order to quantify the influence of a point to the nullity of the cohomology spaces, we introduce a local version of the -cohomology. This provides us with an interesting tool capable to distinguish local features of the quasiconformal geometry of the boundary.
Given a quasi-isometric embedding , where is another hyperbolic simplicial complex with bounded geometry, one can define the pull-back of cochains , see for example [BP03]. For , denote by the collection of all such quasi-isometric embeddings with . We define the space of locally -integrable -cochains of (with respect to ) as
Here and denote the set of -simplices of and respectively. We emphasize the fact that the cochains are defined globally, but the integrability condition is “local”. As we will show in Section 5, is a Fréchet space and the coboundary operators
are Lipschitz continuous.
Consider a point . We define the local -cohomology of with respect to as
The reduced local -cohomology is defined as usual taking the quotient by .
Notice that by definition, the local -cohomology is a quasi-isometry invariant of the pair . More precisely, if is a quasi-isometry between two hyperbolic simplicial complexes as above, then is isomorphic as a topological vector space to . We denote also by the boundary extension.
As for the global cohomology, we can identify the local Orlicz cohomology in degree one with a local Orlicz-Besov space on the parabolic boundary . We suppose that is equipped with an Ahlfors regular parabolic visual metric . We define the local Orlicz-Besov space on as
where is the seminorm defined as in (1.2), but replacing by and integrating over the compact .
Let be a doubling Young function, and be an Ahlfors regular parabolic visual metric on . There exists a canonical isomorphism of Fréchet spaces between and . In particular, is reduced.
These identifications are very useful to define several quasi-isometry invariants. Consider a family of doubling Young functions indexed on a totally ordered set , and suppose that it is non-decreasing in the sense that if in , then (see Section 2 for the definition of the relation ). We define the critical exponent of as
Notice that by Theorem 1.2, there is a canonical inclusion whenever . When the family of Young functions is given by , this exponent is the well known critical exponent associated to the -cohomology of . An analogous exponent can be defined for the local Orlicz cohomology.
It is a unital Banach algebra when equipped with the norm . The spectrum of , denoted by , is a Hausdorff compact topological space invariant by Banach algebra isomorphisms. In particular, the spectrum, as well as its topological dimension , are quasi-isometry invariants of . For instance, given an indexed family as before, the function provides another quasi-isometry invariant of .
The spectrum of is a quotient space of , where the equivalence relation is given by
The -equivalence classes provide a partition of which must be preserved by the boundary homeomorphism of any self quasi-isometry of .
The same considerations can be carried out for the local Orlicz cohomology, by considering the unital Fréchet algebra of continuous Orlicz-Besov functions on the parabolic boundary . In particular, the boundary homeomorphism of a self quasi-isometry of which fixes the point , must preserve the local -cohomology classes of .
1.2. On quasi-isometries of Heintze groups
We will focus on the global and local Orlicz cohomology in degree one of , for the family of doubling Young functions given by
In order to simplify the notation, we indicate the Orlicz spaces and the norms associated to the functions with the superscript “”. The set of pairs is endowed with the lexicographic order, so we obtain a non-decreasing family of Young functions as in the previous section.
We will define a non-decreasing sequence of closed Lie subgroups of ,
whose left cosets will be identified with local cohomology classes for appropriate choices of the parameters and .
Denote by the distinct eigenvalues of , and let be a basis of on which assumes its Jordan canonical form. In the basis , we have the decomposition
Here denotes a Jordan block of size associated to the eigenvalue . Let be the generalized eigenspace associated with .
Let , and for each , let
That is, is the Lie sub-algebra of generated by . Define to be the closed Lie subgroup of whose Lie algebra is . Note that the set of left cosets is a smooth manifold, and the canonical projection is a smooth map.
For each , we also let be the maximal size of the Jordan blocks with eigenvalue . Consider the -eigenvectors belonging to the Jordan blocks of size , and denote by the vector space their span. Finally, consider the Lie subalgebra spanned by , and its corresponding closed Lie subgroup. Notice that only when .
The parabolic boundary can be identified with . A left invariant parabolic visual metric can be defined on , and so that acts as a dilation. From this, one easily checks that is Ahlfors regular (see Section 6 for more details). In the statement of the next theorem we write for
Theorem 1.4 (Local classes with respect to ).
For each , consider the exponent , and set .
Suppose and . The spectrum of is homeomorphic to .
Suppose and .
If , the spectrum of is homeomorphic to .
If , the spectrum of is homeomorphic to .
Suppose , , and . The spectrum of is homeomorphic to .
In particular, in all cases, the -local cohomology classes on coincide with the lefts cosets of the corresponding subgroup.
Let . Then the local critical exponent satisfies
The picture is quite different for the local cohomology with respect to the points of .
Theorem 1.5 (Local cohomology with respect to ).
The local -cohomology of with respect to a point is trivial if, and only if, .
In particular, for any . Theorem 1.1 follows therefore from Theorems 1.4 and 1.5, see Section 6. Notice that in the Carnot type case, for any . As an immediate consequence, we obtain the following result for the global cohomology.
The critical exponent of the -cohomology of is given by
Moreover, the -cohomology is also trivial at this critical exponent.
As an example, consider the Heintze group introduced at the beginning of this section. The critical exponents are in this case is and for any . The subgroup is . Notice that even though the conformal dimension of is equal to , it is not attained. This is proved in [HP11][Thm. 1. 8] by techniques of two dimensional conformal dynamics which do not apply to the higher dimensional case. We refer the reader to [MT10] for an account on conformal dimension.
The -coordinate of the critical exponent can be interpreted as a second order quasi-isometry invariant “dimension” of a hyperbolic complex as above. When the Ahlfors regular conformal dimension of is attained, the critical exponents satisfy the inequality for any , see Lemma 5.2.
If , the Ahlfors regular conformal dimension of is not attained.
The pointed sphere conjecture is not settled in the Carnot type case. Our methods do not apply, essentially, because Carnot groups equipped with Carnot-Carathéodory metrics are Loewner spaces [Hei01]. In particular, they contain “a lot of rectifiable curves”, which makes difficult to distinguish points by invariants strongly related to the moduli of curves.
The restrictions imposed on self quasi-isometries of preserving a foliation at infinity are manifest in the rigidity results due to Xie, in the case when is abelian or isomorphic to a Heisemberg group. It is there shown that self quasi-isometries are almost-isometries, i.e. a -quasi-isometry. This question is motivated by the work of Farb and Mosher on abelian-by-cyclic groups [FM00]. We apply Xie’s approach to our more general context.
Let be a purely real Heintze group, and suppose that the normalizer of , , is strictly bigger than . Then, any self quasi-isometry of is an almost isometry.
The Corollary 1.8 applies, in particular, in the case when is abelian or isomorphic to a Heinsemberg group, and is not of Carnot type, generalizing therefore the previous known results.
We obtain also results regarding the quasi-isometric classification of Heintze groups. It is conjectured that two purely real Heintze groups are quasi-isometric if, and only if, they are isomorphic [Ham87, Cor12]. By [Pan89c, Theorem 2], the conjecture is true when and are both purely real Heintze groups of Carnot type. Notice that if is the restriction of to , then is a Heintze group of Carnot type. Also, that if there exists a quasi-isometry between two Heintze groups and , then there exists a quasi-isometry sending to [Cor12, Lemma 6.D.1]. From Pansu’s theorem and Theorem 1.4, we obtain the following consequence.
Let and be two purely real Heintze groups. If they are quasi-isometric, then and are isomorphic. In particular, if is of Carnot type and is not, then they are not quasi-isometric.
In the abelian type case, that is, when is abelian, the algebra is not playing any role, and we obtain another, and more direct, proof of the following result of Xie.
[Xie14a, Theorem 1.1] Let and be two purely real Heintze groups of abelian type. If they are quasi-isometric, there exists such that and have the same Jordan form. In particular, and are isomorphic.
We refer the reader to [Cor12, Section 6.B] for more details about the quasi-isometric classification of Heintze groups. Notice that Theorem 1.4 provides new invariants related to the sizes of the Jordan blocks of . We were not able to compute the spectrum of for all the possible values of . One could expect to show, by a more refined analysis, that the Jordan form of the derivation , up to scalar multiplication, is a quasi-isometry invariant of the Heintze group, generalizing thus the first conclusion of Corollary 1.10.
1.3. Notations and conventions
To make the notation clearer we will write the constants in sans-serif font, e.g. , , etc.. If and are non-negative real functions defined on a set , we say that if there exists a constant , such that for all . If both inequalities are true, and , we say that and are comparable and we denote it by .
I would like to specially thanks Pierre Pansu for all his help and advice during the realization of this work, and for explaining me his works on -cohomology. I am also particularly indebted to Romain Tessera for sharing with me his ideas about Orlicz spaces and their applications to Heintze groups. I also thanks Yves de Cornulier for helpful discussions on quasi-isometries of Heintze groups. This work was supported by the ANR project “GDSous/GSG” no. 12-BS01-0003-01.
2. Preliminaries on the theory of Orlicz spaces
In this section we recall some basic facts about Orlicz spaces which will be used throughout this articles. We refer to [RR91] for a general treatment. Recall that a Young function is an even convex function which satisfies , and . Note that we require to be finite valued, so in particular, is locally Lipschitz. We will also assume that if .
Any Young function can be represented as an integral
where is nondecreasing left continuous and [RR91, Cor. 2 Ch. 1]. The function coincides with the derivative of except perhaps for at most a countable number of points.
Let the space be given with a -algebra and a -finite measure . For any measurable function on , the Luxembourg norm of is defined as
where it is understood that . The Orlicz space is the vector space of measurable functions on such that [RR91, Thm. 10 Ch. 3]. Up to almost everywhere null functions, is a Banach space with the norm . As usual, we simply write when is the counting measure.
If is any constant, the identity map is continuous and bijective, and therefore, by the open mapping theorem, and are equivalent norms:
Let us point out that when the measure is finite, the Orlicz space is contained in and the inclusion is continuous [RR91, Cor. 3 Ch. 1]. In particular, in locally compact spaces equipped with a regular measure, Orlicz integrable functions are locally integrable.
We will not need the Hölder inequality in this paper, but let us remark that it holds exactly as in the ordinary case by considering the convex conjugate of [RR91, Thm. 3 Ch. 1]. This inequality serves also to define the Orlicz spaces by means of the so called Orlicz norm instead of the Luxembourg norm.
The main difficulty in dealing with the norm is that it is not comparable to the function
This is only the case when is equivalent to an ordinary power function , . Nevertheless, we will be able to avoid this difficulty by applying Jensen’s inequality in order to exchange and an integral or a sum symbol.
2.2. Decay conditions
For the Orlicz spaces considered here, only the decay properties of for small values of will be relevant for us. Roughly speaking, we do not want our Young functions to be too small near zero. This is precisely the meaning of Definition 1.1, and we will mainly work in this article with doubling Young functions.
Here are some important features of Orlicz spaces of doubling Young functions. First, notice that since is non-decreasing, we have
A useful way to check the doubling property is by considering the exponent
Then, is doubling if and only if [RR91, Cor. 4 Ch. 2]. It is important to note that when is doubling, then
see [RR91, Thm. 2 Ch. 3]. Moreover, when is doubling, if is a sequence in , then
see [RR91, Thm. 12, Ch. 3]. In particular, when is a Radon (resp. volume) measure on a locally compact Hausdorff space (resp. Riemannian manifold) , then the usual approximation by continuous (resp. smooth) functions in the norm holds.
A doubling Young function has polynomial decay with exponent for any . That is, there exist , and , such that . In general, the converse is not true. We also remark that when is doubling of constant , then for all .
All these conditions are imposed for small values of . When dealing with discrete Orlicz spaces, the behavior of at infinity is essentially irrelevant. To justify this, let us introduce the following equivalence relation among Young functions.
Define if there are constants and , so that for . We say that for small , if and .
Suppose that is a countable set, and let , , be a pair of equivalent Young functions for small . Then the norms and are equivalent.
The following lemma will be used later in Section 4.
Let be a doubling Young function with constants and . Then for all and we have .
For each , define the function by . Notice that because and is increasing. Since for all , we have
That is, . This finishes the proof. ∎
3. Orlicz cohomology and quasi-isometry invariance
In this section, we extend the classical notion of -cohomology by considering the larger class of Orlicz spaces. It can be defined as a simplicial, coarse, or De Rham cohomology, depending on the structure of the space under consideration. For our later applications in Section 6, it will be more natural to work with differential forms. Nevertheless, the quasi-isometry invariance is easier to prove in the simplicial context.
In contrast to the case, the identification of the discrete and continuous cohomologies is less clear for a general Young function. We will prove it for the degree one cohomology, by using the coarse definition as an intermediary (see Section 3.2).
3.1. The simplicial Orlicz chomology
We first prove the quasi-isometry invariance of the simplicial Orlicz cohomology. It is remarkable that this result is true for any Young function , and therefore, in great generality. The proof follows the same lines as the classical proof for the case, see [BP03].
Let be a finite dimensional, uniformly contractible, simplicial complex with bounded geometry, see the Introduction for the definition. We denote the geodesic distance on by . For , let be the set of -simplices of , and let be the vector space of -chains on , i.e. finite real linear combinations of the elements in .
We will use the following notations: if is a chain in , the length of , denoted by , is the number of simplices in , , and . Note that our assumptions imply that there exists a function such that any ball of radius contains at most simplices. We write for short.
Let be a Young function. The -th space of -integrable cochains of is by definition the Banach space ; i.e. the space of functions such that
The standard coboundary operator is defined by duality: for and , let .
By the convexity of , and the bounded geometry assumption, is well defined and is a bounded operator. The proof relies on the following argument which will be used several times in this article. Let us explain it in detail.
Let and , and suppose that . Note that for any , we have by convexity
Summing over , we obtain
By (2.1), . In particular, the set of such that the sum on the right hand side is bounded from above by is not empty, and we can take arbitrarily close to . For such an , we have
which implies . Taking infimum, we get . Therefore, , where the multiplicative constant depends only on and .
The -th -cohomology space of is by definition the topological vector space
The reduced cohomology is defined by taking the quotient by ; it is a Banach space.
We focus now on the quasi-isometry invariance of the -cohomology spaces. Recall that a function between two metric spaces is:
quasi-Lipschitz if there exist constants and such that
uniformly proper if there exists a function such that for any ball in , we have .
a quasi-isometry if it is quasi-Lipschitz and there exists a quasi-Lipschitz function such that and are at bounded distance from the identity.
We can now state the main result of this section.
Theorem 3.1 (Quasi-isometry invariance).
Let be any Young function, and , be two uniformly contractible simplicial complexes with bounded geometry.
Any quasi-Lipschitz uniformly proper function induces continuous linear maps .
If are two quasi-Lipschitz uniformly proper functions at bounded distance, then .
If is a quasi-isometry, then is an isomorphism of topological vector spaces.
The same statements hold for the reduced -cohomology.
The proof relies on the following two key lemmas which are proved in [BP03, Section 1]. They serve to define pull-backs and homotopies from quasi-isometries.
Lemma 3.2 ([Bp03]).
Let and be two uniformly contractible simplicial complexes with bounded geometry. Any quasi-Lipschitz uniformly proper function induces maps verifying the following conditions:
commutes with the boundary operator, i.e. , and
for each , there are constants and , depending only on and the geometric data of and , such that
The map is constructed by induction on . For and , is defined to be any vertex of at uniformly bounded distance from . The induction can be carried out since is uniformly contractible.
Lemma 3.3 ([Bp03]).
Let be two quasi-Lipschitz uniformly proper functions at bounded uniform distance. Then there exists a homotopy between and . That is, a map verifying
for , , and
for , , .
As before, and are uniformly bounded on by constants and , depending only on the geometric data of and .
The homotopy is also defined by induction on . For and , is defined to be any -chain in verifying
Proof of Theorem 3.1.
Let us prove (1). For , define the pull-back
We show that belongs to . The proof will also show the continuity of . For , by convexity of , we have
For fixed , the uniform properness of implies that the set of for which , is contained in a ball of of radius . Then, their number is bounded by . This implies
As in (3.2), this gives
Denote by and the coboundary operators of and respectively. Then , since commutes with the boundary operator. Thus, induces a continuous linear map . This also holds for the reduced cohomology by the continuity of .
Let us prove (2). Suppose that . For , define the pull-back