Topology of automorphism groups of parabolic geometries
C. Frances, K. Melnick

TL;DR
This paper proves that the automorphism group of any parabolic geometry can be given a Lie group structure where the $C^0$ and $C^{inite}$ topologies coincide, and it is closed within the homeomorphism group of the manifold.
Contribution
It establishes the Lie group structure of automorphism groups of parabolic geometries and their topological properties, unifying their $C^0$ and $C^{inite}$ topologies.
Findings
Automorphism group admits a Lie group structure.
The $C^0$ and $C^{inite}$ topologies coincide.
Automorphism group is closed in the homeomorphism group.
Abstract
We prove for the automorphism group of an arbitrary parabolic geometry that the and topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.
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Topology of automorphism groups of parabolic geometries
Charles Frances and Karin Melnick
Abstract.
We prove for the automorphism group of an arbitrary parabolic geometry that the and topologies coincide, and the group admits the structure of a Lie group in this topology. We further show that this automorphism group is closed in the homeomorphism group of the underlying manifold.
This project was initiated during a visit of the second author as Professeur Invitée in May 2016 to the Université de Strasbourg, whom the authors thank for this support. Melnick partially supported by NSF grant DMS 1255462.
1. Introduction
It is well known that the automorphism group of a rigid geometric structure is a Lie group. In fact, as there are multiple notions of rigid geometric structures, such as -structures of finite type, Gromov rigid geometric structures, or Cartan geometries, the property that the local automorphisms form a Lie pseudogroup is sometimes taken as an informal definition of rigidity for a geometric structure.
There remains, however, some ambiguity about the topology in which this transformation group is Lie. It is a subgroup of , assuming the underlying structure is smooth, so one may ask whether it admits the structure of a Lie group in the , for some positive integer , or even the compact-open, topology. A related interesting question is whether the automorphism group is closed in .
Theorems of Ruh [14] and Sternberg [17, Cor VII.4.2] state that, if is the automorphism group of a -structure of finite type of order , then is a Lie group in the topology on . Gromov proved a similar result in [5, Cor 1.5.B] for a smooth Gromov--rigid geometric structure. In the case of a smooth Riemannian metric , the results above yield a Lie group structure for the -topology on the isometry group .
The classical theorems of Myers and Steenrod [11], however, say that in this Riemannian case the and topologies coincide on for all . Nomizu [12] proved the same for the group of affine transformations of a connection (under an assumption of geodesic completeness, which can be removed). The essence of the proof is that exponential coordinates locally convert affine transformations to linear maps, and a sequence of linear transformations converging automatically converges .
This article is concerned with the topology of local automorphisms of parabolic geometries (see section 1.2 below for the general definition). These form a rich class of differential-geometric structures which behave differently from Riemannian metrics in the sense that their automorphisms can have strong dynamics, so, for example, a convergent sequence of automorphisms need not limit to a homeomorphism. Parabolic geometries do not determine a connection; without the exponential map, it is no longer clear that a -limit of smooth automorphisms should be smooth.
1.1. Statement of main results
We first briefly survey some results for specific parabolic geometries, which will be generalized by our main theorem. We remark that the first two theorems below, of Ferrand and Schoen, are proved by geometric-analytic techniques that are quite specific to the structures in question.
- •
In the course of proving the Lichnerowicz Conjecture on Riemannian conformal automorphism groups, Ferrand showed, using techniques of quasiconformal analysis, that if a homeomorphism is a limit of smooth conformal maps, then is also smooth and conformal [1, 9].
- •
Schoen [15] reproved Ferrand’s result above, and extended it to strictly pseudoconvex -structures. His proof uses scalar curvature and the conformal Laplace operator in the conformal case, and the analogous Webster scalar curvature and pseudoconformal subelliptic operator in the CR setting.
- •
In [3], the first author proved for conformal pseudo-Riemannian structures that if a sequence of smooth local conformal transformations converges , then it converges . His approach is very different from the analytic techniques of [1] and [15]: he uses the Cartan connection associated to these structures and the dynamics of the action on null geodesics.
We prove a generalization of the results recounted above to local automorphisms of arbitrary parabolic geometries. Parabolic geometries are a broad family of geometric structures which nonetheless admit an extensive general theory. Well known examples include the conformal semi-Riemannian structures and strictly pseudoconvex CR structures mentioned above, as well as more general nondegenerate CR structures, projective structures, and so-called path geometries, which encode ODEs (see [19] for a comprehensive reference). Definitions 1.4 and 1.5 below explain precisely what is meant by parabolic geometry and automorphism/automorphic immersion. An automorphic immersion can be informally defined as a differentiable immersion , where is an open set, which preserves the Cartan geometry on . When and is also a diffeomorphism, then is said to be an automorphism of . The set of automorphisms is a group that will be denoted . Our main results can then be stated as follows:
Theorem 1.1**.**
Let be a smooth parabolic geometry. Let be a sequence of automorphic immersions of converging in the topology on to a map . Then is smooth and also in the topology.
In section 3.3 we will also prove the following:
Theorem 1.2**.**
Let be a smooth parabolic geometry. Then is a Lie transformation group in the compact-open topology. Moreover, is closed in for this topology.
1.2. Definitions
Parabolic geometries are most conveniently defined in terms of Cartan geometries. Let be a Lie group with Lie algebra , and a closed subgroup. We will assume throughout the article that the pair is effective, meaning acts faithfully on . A noneffective pair can always be replaced by an effective one, with the same quotient space (see [16]).
Definition 1.3**.**
A Cartan geometry on a manifold , with model space comprises , where is a principal -bundle, and is a -valued one-form on satisfying:
- •
For all , is a linear isomorphism.
- •
For all , , where denotes the right translation by on .
- •
For all , , where .
The basic example of a Cartan geometry modeled on is the flat geometry on comprising , where is the Maurer-Cartan form.
Definition 1.4**.**
*A parabolic geometry is a Cartan geometry modeled on , where is a semisimple Lie group with finite center and without compact local factors, and is a parabolic subgroup. *
Our notion of parabolic subgroup is the standard one, which will be recalled in Section 2.5.1.
Essentially all classical rigid geometric structures correspond to a canonical Cartan geometry. The process of canonically associating a Cartan geometry is called the equivalence problem for a given geometric structure (see [16] for examples). Parabolic geometries admit a uniform solution of the equivalence problem, in which each corresponds to a type of “filtered manifold” (barring one exception, projective structures); see [19, Sec 3.1], [18].
Definition 1.5**.**
For a smooth Cartan geometry with , an automorphism is which lifts to a bundle automorphism of satisfying . The group of automorphisms is denoted .
For an open subset , a smooth immersion is an automorphic immersion of if it lifts to a bundle map satisfying .
As is effective, the elements correspond bijectively to their lifts to satisfying , and similarly for automorphic immersions (see [10, Prop. 3.6]).
1.3. Lie topology on the automorphism group
For a smooth Cartan geometry on , the group can be endowed with the structure of a Lie transformation group as follows (we refer to the definition in [13, Chap. IV] of Lie transformation group). The Cartan connection defines a framing of , the pullback by of any basis in . The automorphisms of a framing form a Lie transformation group; more precisely:
Theorem 1.6**.**
*(S. Kobayashi [8, Thm I.3.2])
Let be a smooth, connected manifold with a smooth framing .*
- (1)
* admits the structure of a Lie transformation group.* 2. (2)
For , the -topology on coincides with the Lie topology. 3. (3)
A sequence converges in the Lie topology if and only if there exists such that converges in .
Denote by the group of bundle automorphisms of preserving . This is a -closed subgroup of , so it is closed in the Lie topology and inherits the structure of a Lie transformation group. The isomorphism then provides the latter with the structure of a Lie group, in fact of a Lie transformation group of . The underlying topology on , the pullback of the topology on , will henceforth be referred to as the Lie topology. For , the automorphic immersions defined on admit a similarly defined topology, which we will also call the Lie topology.
Recall that the Lie topology on , as well as all -topologies, are second countable. A sequence of automorphic immersions of converges in the Lie topology if and only if the lifted sequence converges . Thus if converges for the Lie topology to an automorphic immersion, then it does for the -topology on . In cases where is a subbundle of the -frames of , and are the corresponding natural lifts of , then convergence of on conversely implies convergence in the Lie topology. Such is the case for many parabolic geometries, but this property in general is unclear. Our proofs will go via the Lie topology on , thus showing that it coincides with all -topologies, , and similarly for automorphic immersions of .
1.4. Structure of the Proof
A sequence of automorphic immersions converging in the topology gives rise to a holonomy sequence in . The action of on reflects many features of the action of on . Section 2 contains the definition of holonomy sequences and their equicontinuity properties relative to those of . In Section 3, we translate the problem to a statement about holonomy sequences on . The proof of this statement, Theorem 3.1, proceeds by induction on . The base case, , is recalled from [2] in Section 4. The task for the remainder of the paper is, given a holonomy sequence not conforming to the conclusion of Theorem 3.1, to find an invariant lower-rank subvariety of on which exhibits the same behavior, thus contradicting the induction hypothesis. Section 5 develops tools for identifying such a lower-rank subvariety, corresponding to certain manipulations on the root spaces of . In Section 6, we apply these tools to complete the induction step.
2. Holonomy and equicontinuity with respect to segments
Let be a Cartan geometry modeled on , not necessarily parabolic.
Definition 2.1**.**
A sequence of automorphic immersions of is equicontinuous at if there exists such that for any in , the sequence .
If converges , then is clearly equicontinuous at every point of . The following theorem says that conversely, equicontinuity at a single point implies local -convergence, at least for parabolic geometries.
Theorem 2.2**.**
Let be a smooth parabolic geometry and a sequence of automorphic immersions equicontinuous at . Then there exists an open neighborhood of on which a subsequence of converges to a smooth map .
Note that Theorem 2.2 implies Theorem 1.1.
2.1. Holonomy sequences
Let be a sequence of automorphic immersions of which is equicontinuous at , with lifts . Associated to is a holonomy sequence in , whose behavior around the base point reflects much of the local behavior of around .
Definition 2.3**.**
Let in . A sequence of is a holonomy sequence of along when there exist such that and are bounded in . A holonomy sequence of at is any holonomy sequence along some sequence .
We will denote by the set of all holonomy sequences of at . Equicontinuity of at ensures that is nonempty. Indeed, given such that , choose any and . Then there exists a sequence in such that , so .
2.2. Equicontinuity with respect to segments
Equicontinuity of a sequence at will have strong consequences on the local behavior of its holonomy sequences around the base point . A useful notion to capture this local behavior is equicontinuity with respect to segments. An unparametrized segment in is a set of the form , for some . Remark that distinct may define the same unparametrized segment.
We fix a Riemannian metric in a fixed neighborhood of in , with respect to which we will measure the length of segments in this neighborhood, and denote the results by .
Definition 2.4**.**
*A sequence in is equicontinuous with respect to segments if when a sequence of segments satisfies , and , then every cluster value of in is in . *
Observe that the condition , hence the very notion of equicontinuity with respect to segments, does not depend on the choice of Riemannian metric, since any two are bi-Lipschitz equivalent in a neighborhood of .
2.3. Relation of equicontinuity and equicontinuity with respect to segments
Proposition 2.5**.**
Let be a Cartan geometry and a sequence of automorphic immersions of . If is equicontinuous at , then every holonomy sequence is equicontinuous with respect to segments.
The proof will use the development of curves , a notion which we now recall. Given such a smooth curve , the equation , where is the Maurer Cartan form of , defines an ODE on . The solution such that will be called the development of .
The Cartan connection also yields an exponential map on : any in defines the -constant vector field on by ; denote the corresponding local flow. Observe that whenever , the flow is globally defined and corresponds to right multiplication by in the bundle (by the third axiom in Definition 1.3). The exponential map at is defined in a neighborhood of the origin in by
[TABLE]
Shrinking if necessary makes the exponential map at a diffeomorphism onto a neighborhood of in . For , we will denote the exponential of at in by , and the exponential in the Lie group by .
It is easy to see that whenever is the lift of an automorphic immersion of , then
[TABLE]
The -equivariance property of leads to a corresponding equivariance property for the exponential map for all
[TABLE]
Last, we recall the following crucial reparametrization lemma.
Lemma 2.6** ([4], Proposition 4.3).**
Let be smooth curves, with , and let be a smooth map satisfying .
- (1)
Assume that for the developments and , the relation holds in for every . Then holds in . 2. (2)
In particular, if , and if there exists a smooth , with and , such that
[TABLE]
then, for every such that or is defined,
[TABLE]
**Proof: **(of Proposition 2.5) Assume for a contradiction that is equicontinuous at , but that some holonomy sequence of at does not act equicontinuously with respect to segments. Then is bounded for a bounded sequence projecting to . After passing to a subsequence, we can assume and .
Since is not equicontinuous with respect to segments, passing again to a subsequence, there exists a sequence of segments , with , as well as a sequence in converging to , such that for all :
[TABLE]
This condition can be expressed by the relation, valid for all :
[TABLE]
Here, denotes a smooth path with and a nondecreasing diffeomorphism. Given arbitrary small, let be such that for all . Then write
[TABLE]
Note that . Thus for sufficiently small, we can replace and by and , so that (2) holds, with the extra property that is defined for all , and is in an injectivity domain of the map . In particular, if we call , the fact that implies, shrinking again if necessary, .
The next step is to show that is defined for large enough, and converges to . To this aim, define a left-invariant Riemannian metric on by left translating any scalar product on , and a corresponding Riemannian metric on , with
[TABLE]
By the definition of , if is a curve in and its development in , then . Fix small enough that , the -ball of center and radius has compact closure in .
Now consider the curve . We fix a small submanifold of containing , which is transverse to the fibers of , and such that the restriction of to yields a diffeomorphism , where is a neighborhood of in . For large enough, there exists a smooth , with , such that is contained in . Of course . Two Riemannian metrics on are always locally bi-Lipschitz equivalent, hence there exist such that for large enough:
[TABLE]
We infer that ; in particular, for , . Now consider, for each , the first-order ODE on :
[TABLE]
with initial condition . If , is a maximal interval of definition for , then for all , , , is a maximal solution of our ODE, by Lemma 2.6. By the definition of , we have . If , the inequality implies that is included in the relatively compact set ; this contradicts the maximality of . We thus infer , which ensures that , hence is defined. Moreover, , so . Projecting to gives .
Now Lemma 2.6, combined with equation (3) above says that for all ,
[TABLE]
Projecting this relation on , we obtain
[TABLE]
After possibly passing to a subsequence, the right-hand term converges to , while we just showed ; this yields the desired contradiction with the equicontinuity of at .
2.4. Vertical and transverse perturbations of holonomy sequences
Proposition 2.5 translates equicontinuity of at to a property of sequences in , which are in turn sequences of acting on . In this section we define several operations on sequences in which preserve .
Holonomy sequences involve many choices: of , of , and of , in the notation of Definition 2.3. The right and left vertical perturbations of correspond to other possible choices of and , respectively.
Definition 2.7**.**
Let be a sequence in . A vertical perturbation of is a sequence where and are two bounded sequences in .
Transverse perturbations of correspond roughly to other possible choices of converging to .
Definition 2.8**.**
For a sequence of , a sequence of is said to be a transverse perturbation of when there exist two sequences and in such that:
- (1)
** 2. (2)
The sequences and both converge to [math]. 3. (3)
For every , belongs to .
The other choice of in this case is , as will be seen in the proof below.
Lemma 2.9**.**
Let be a Cartan geometry, and let be a sequence of automorphic immersions. For any , the set of holonomy sequences is stable by vertical and transverse perturbations.
**Proof: ** We consider a sequence belonging to . By definition, there exists a bounded sequence in such that is bounded, and the projection on converges to .
Assume that is obtained from by vertical perturbation, namely there exist bounded sequences and in such that . Then is bounded in , and still projects on . Moreover
[TABLE]
is still bounded in . It follows that is a holonomy sequence at .
We now handle the case of a transverse perturbation . The sequence is bounded and , hence is bounded in , too; moreover, converges to . It remains to show that is bounded in . Write this expression as . By the equivariance (1) of the exponential map,
[TABLE]
Point in the definition of transverse perturbation says that belongs to for all . Thus
[TABLE]
where is a smooth path in passing through when . Lemma 2.6 then implies
[TABLE]
Right translation by gives . This expression is bounded, because is a bounded sequence, and tends to zero by definition of a transverse perturbation.
2.5. Admissible operations
In this section, we specialize to a parabolic model space, and define some operations on holonomy sequences specific to parabolic geometries. We first introduce some notation in .
2.5.1. Notation in
Let be semisimple with no compact local factors and with finite center. We denote by a Cartan involution of the semisimple Lie algebra . Associated to , we choose a Cartan subspace , and a set of simple roots. The positive and negative roots are denoted and , respectively. The usual decomposition of the Lie algebra into root spaces is
[TABLE]
Recall that the Lie algebra is centralized by , and lies in the Lie algebra comprising the -eigenspace of the Cartan involution .
We will denote by (resp. ) the sum (resp. ).
The minimal parabolic subalgebra of is . A general parabolic subalgebra is one containing , and is obtained as follows (up to conjugacy in ): there exists , possibly empty, such that
[TABLE]
where is the set of roots in which are in the span of . A parabolic subgroup of is any Lie subgroup with Lie algebra , for some . We will sometimes denote this group simply when is understood.
We denote by the nilpotent radical of , which equals . Here stands for the positive roots written as linear combinations of roots in involving at least one root which is not in . Notice that is an ideal of and of . Finally, we call the Lie algebra .
We denote by , and the connected Lie subgroups of with Lie algebras , and , respectively; they are all subgroups of .
2.5.2. Reduced holonomy sequences
A sequence in will be called reduced when it is a sequence of .
Lemma 2.10**.**
Any sequence in can be converted by left and right vertical perturbation to .
**Proof: ** Consider the Levi decomposition of , where is the connected reductive subgroup of with Lie algebra spanned by and the positive and negative root spaces of . Write according to this decomposition. As is reductive, it admits a decomposition, according to which , with and . As has finite center, is contained in a maximal compact subgroup of and is a maximal compact subgroup of . Then , where . Now is the desired reduced sequence.
2.5.3. Weyl reflections
For parabolic, these are transformations of holonomy sequences in , which will be useful in our proof.
For any root , the Weyl reflection is with
[TABLE]
Recall that for positive, preserves and , assuming is not a root (in which case, preserves and ). Recall that whenever is a root, then is an integer.
For any root , there exists , such that preserves , and the action of on coincides with that of (see [7, Prop 6.52c]). In the sequel, we will denote by any automorphism of such that the action induced on preserves and sends every root space to the corresponding ; for instance, could be conjugacy by .
Let . If a root is a linear combination with integer coefficients of roots in , then so is ; thus preserves . As sends all positive roots except multiples of to positive roots, it also preserves . We conclude that for every , an automorphism preserves the connected subgroups , , and the identity component ; in particular, it sends sequences in to in . Note that in general, may not be invariant by .
2.5.4. Definition of admissible operations, perturbations
Definition 2.11**.**
Let be a parabolic variety with . For a sequence of , an elementary admissible operation on is of one of the three following types:
- (1)
A vertical perturbation of . 2. (2)
A transverse perturbation of . 3. (3)
For in , a Weyl reflection applied to , with .
An admissible perturbation of a sequence in is a sequence which is obtained from by finitely many elementary admissible operations.
Note that the result of an admissible perturbation of a sequence of is always in . Weyl reflections are only allowed on sequences of , which must be kept in mind when applying successive admissible operations.
We conclude this section with an important remark about Weyl reflections. We observed at the end of the last paragraph that a Weyl reflection always coincides with the conjugacy by some element . We also observed that preserves the identity component of , so that actually belongs to , the normalizer of in . This normalizer has Lie algebra (see [19, Lemma 3.1.3, Cor. 3.2.1(4)]), so that the inclusion holds. Observe that in general, these groups need not coincide. However, when , any Weyl reflection is actually a vertical perturbation of . We thus get a straightforward rephrasing of Lemma 2.9, namely
Lemma 2.12**.**
Let be a parabolic geometry modeled on , where . Let , and let be a sequence of automorphic immersions which is equicontinuous at . Then if is in , any admissible perturbation of is in .
The case of equality will thus be technically more convenient, since it means that Weyl reflections on holonomy sequences again yield holonomy sequences. It is explained in Section 3.2 why this equality may be assumed.
3. Translation of the main theorem to the model space
Via the holonomy sequences associated to an equicontinuous sequence of automorphic immersions, we can translate Theorem 2.2 to an assertion about sequences of acting equicontinuously with respect to segments on .
Theorem 3.1**.**
Let be a parabolic variety with . Given a sequence of which, together with all of its admissible perturbations, acts equicontinuously with respect to segments on , the factor is bounded.
Theorem 3.1 is proved in sections 4, 5 and 6.
3.1. Derivation of Theorem 2.2 from Theorem 3.1
Given a sequence of automorphic immersions as in the statement of Theorem 2.2, let be a holonomy sequence of at . We can assume by Lemmas 2.9 and 2.10 that for all .
We will first deduce Theorem 2.2 under the extra assumption that equals . Section 3.2 explains how to dispense with this assumption.
Proposition 2.5 ensures that acts equicontinuously with respect to segments on . Lemma 2.12 says that in fact every admissible perturbation of does (under our assumption ). Now the hypotheses of Theorem 3.1 are satisfied. The conclusion implies that is a right vertical perturbation of , which by Lemma 2.9 also belongs to . The action of on preserves the subalgebra ; denote by the endomorphism .
Lemma 3.2**.**
The sequence is bounded in .
**Proof: ** The representation of on is diagonalizable with eigenvalues . Assume for a contradiction that is unbounded; we may assume that is unbounded, and after passing to a subsequence, that . Taking a subsequence also allows us to assume that in , the sequence converges to .
For each , let be in the -eigenspace of such that ; these can moreover be chosen in the injectivity domain of , and such that is in the injectivity domain of . Set . Because , the exponential is defined for sufficiently large , and satisfies
[TABLE]
Projecting to gives a contradiction to the equicontinuity of at : , while .
Now again passing to a subsequence of , we may assume that tends to some . Let be a compact set containing both sequences and , and let and be relatively compact neighborhoods of [math] in , such that:
- (1)
for every . 2. (2)
For every , the map is defined on and , and is a diffeomorphism from and onto their respective images.
There exists an open neighborhood of , such that for close enough to . Then define the smooth map by . Because converges smoothly to , and since on , for large enough,
[TABLE]
converges smoothly to on . Thus Theorem 2.2 is proved.
3.2. Justification of the assumption
Let be a sequence of automorphic immersions as in Theorem 2.2. In general , and they have the same Lie algebra, as remarked above (again, see [19, Lemma 3.1.3, Cor. 3.2.1(4)]). Thus is an isogenous supergroup of . The following lemma gives a general procedure for inducing a Cartan geometry modeled on to one modeled on , with respect to which the automorphism group behaves nicely.
Lemma 3.3**.**
Let be a Cartan geometry on the manifold , modeled on . Let be a closed subgroup, with and . Then there exists a Cartan geometry on the manifold , modeled on , such that:
- (1)
Every automorphic immersion of is an automorphic immersion of . 2. (2)
The corresponding inclusion of into is a homeomorphism onto a closed subgroup with respect to the Lie topologies on each.
**Proof: ** The bundle is obtained as the quotient , where acts diagonally by , freely and properly. There is an obvious commuting right -action on , which descends to , making it a -principal bundle over .
To construct the Cartan connection on , we first build a one-form . For , let
[TABLE]
where is the Maurer-Cartan form of . It is readily checked that satisfies the equivariance relation for every , and that it is invariant under the diagonal action of on . Moreover
[TABLE]
showing that is onto at each point.
For , let be as in Definition 1.3, and let be the curve
[TABLE]
Then
[TABLE]
since . Hence the kernel of contains the tangent space to the -orbits on ; by a dimension argument, these spaces are equal. We infer that induces a -form , which is the desired Cartan connection on .
We prove point (1) for . The argument for automorphic immersions is similar. Let be the lift of to , and define by . The -equivariance of gives the equivariance relation ; obviously, for every . Thus induces a bundle morphism of covering .
To prove that , it remains to check that preserves . To this end, we compute and show that it coincides with :
[TABLE]
but because . Finally,
[TABLE]
as desired, so (1) is proved.
There is a natural -equivariant, proper embedding defined by , the -orbit in of . For with respective lifts and to and , we have .
Now consider a sequence converging for the Lie topology of to an automorphism . By Kobayashi’s theorem (Thm 1.6), the sequence of lifts converges in the -topology of to a diffeomorphism , which clearly preserves . Properness of implies that is closed. Then preserves , because every does. Thus converges smoothly on to , which preserves and covers . It follows that , and by Kobayashi’s theorem, in the Lie topology of . We conclude moreover that is closed in the Lie topology of .
Conversely, given in the Lie topology of , with , the lifts smoothly on . These correspond, as in the proof of (1), to automorphisms and of with on . For any , there exists such that . It follows by Theorem 1.6 (3) that smoothly on each connected component of ; in other words, holds in the Lie topology of . Thus is a homeomorphism onto its image with respect to the Lie topologies on each group.
Now, given a sequence as in Theorem 2.2, Lemma 3.3 with allows us to consider as a sequence of automorphic immersions of , modeled on . The proof of Section 3.1 says that converges smoothly on to a smooth map . We have thus shown that Theorem 3.1 implies Theorem 2.2.
3.3. Derivation of Theorem 1.2
Let converge to in the topology. The aim is to show that , and in the Lie topology on .
By Lemma 3.3 point (2), we may assume that the model space satisfies . As in Section 3.1, admits a holonomy sequence at any , such that is bounded in . Moreover, in the notation of Section 3.1, there is a neighborhood of such that for any accumulation point of in , a subsequence of converges to on . Then , so . Because is a homeomorphism, is injective around [math], hence . As a consequence, converges in .
Now we have with also converging, so tends to some point . As , for sufficiently large , , with in . Now , so . By Theorem 1.6 (3), and the inverses both converge on to smooth maps and , which obviously satisfy . It is easy to see that is a bundle automorphism of preserving . It lifts , hence . Finally, because smoothly on , Theorem 1.6 (2) gives that in the Lie topology on .
4. Proof of Theorem 3.1 in rank one
Our proof of Theorem 3.1 will proceed by induction on . The essential arguments for the base case, , are in the paper [2] by the first author. For the convenience of the reader, the proof is presented here in a manner consistent with our terminology and notation. Theorem 3.1 in this rank one case will actually be a consequence of the following proposition.
Proposition 4.1**.**
Let be a parabolic space, with . If is a sequence of such that is unbounded, then does not act equicontinuously with respect to segments.
Recall the notation of Section 2.5.1. The rank one Lie algebra can be decomposed as a vector space direct sum of subalgebras . The Lie algebra (resp. ) is abelian if , and nilpotent of index , with center of respective dimension , and if is , or . In all cases, (resp. ) will denote the center of (resp. ). The nonequicontinuity will be observed on a restricted class of segments, namely those with
[TABLE]
This set of segments will be denoted and corresponds to conformal circles when , and to chains and their generalizations in the other rank one models. We will adopt the notation (resp. ) for (resp. ).
We now recall two results from [2] regarding these distinguished segments.
Lemma 4.2** ([2], Lemme 2).**
Let be a sequence of segments in . If tends to for the Hausdorff topology, then .
Lemma 4.3** ([2], Proposition 1, (ii)).**
There exists a continuous section . In other words, if a sequence of segments tends to a segment , there is a convergent sequence in such that .
By these two lemmas, if we can find a sequence of segments in tending to , such that tends to (maybe considering a subsequence), then does not act equicontinuously with respect to segments.
The group has exactly two fixed points on , namely and another point . To better understand the action of on , it is convenient to work in the chart , given by . In this chart, elements of act as affine transformations, and segments coincide with half-lines , where and is a unit vector in (for any given norm in which is invariant by the Cartan involution). More precisely, the action of in the chart is linear, and is equivalent to the adjoint action on , and the action of an element , , is given, by the Baker-Campbell-Hausdorff formula, by , .
Now, let us write . By assumption, is an unbounded sequence in . We claim there is an unbounded sequence in such that
[TABLE]
To see this, decompose as a direct sum (observe that when ). Split Equation (5) into two equations in and , namely
[TABLE]
where and are the components of and on , respectively, and
[TABLE]
where and are the components of and on . If is unbounded, then so is , and the same is true for . If is bounded, then is unbounded because is unbounded. This forces to be unbounded.
We can now conclude the proof of Proposition 4.1. Since , then for of norm in , the sequence of half-lines is mapped to by . Now, after taking a subsequence, tends to . Thus for , the sequence of half-lines goes to infinity in the chart , which means that the corresponding sequence of segments tends to in . On the other hand, is equal to a constant segment , and the non equicontinuity of with respect to segments follows.
5. Tools for the induction step: sliding along root spaces
The proof in the previous section for relies heavily on the fact that the action of on the complement of its fixed point is by affine transformations. In higher rank, the -action on is a compactification of an affine action, but no longer a one point compactification. This difference creates significantly more complexity, which motivates our choice to prove Theorem 3.1 by induction rather than directly in arbitrary rank.
The tools developed in this section build on those of Sections 2.4 and 2.5, with the purpose of simplifying holonomy sequences.
5.1. Essential range of
The group exponential of restricts to a diffeomorphism of onto by definition. Moreover, is unipotent, and , so is simply connected; thus restricts to a diffeomorphism .
Fix an ordering of , and endow with the lexicographical ordering. Then we obtain exponential coordinates on and , where is a vector in , on .
Proposition 5.1**.**
Let with exponential coordinates . Then up to vertical perturbation of , we may assume each component sequence is either trivial or unbounded.
**Proof: ** The group is nilpotent; write the lower central series
[TABLE]
Each is abelian and can be spanned by a direct sum of certain root spaces; denote the corresponding set of roots by . Let be the set of roots with bounded. We first multiply on the right by for all , in any order. The Baker-Campbell-Hausdorff formula implies that the resulting exponential coordinates are trivial or bounded for all . Then proceed sequentially through for to obtain satisfying the conclusion of the proposition.
We remark that can also be assumed trivial or bounded by a similar argument, which is not given because this fact is not needed below.
Definition 5.2**.**
Let with exponential coordinates . The essential range of , denoted , is the set of roots for which the component sequence is unbounded.
5.2. Transverse and vertical sliding along root spaces
In our proof by induction on the rank of , the goal will be, given a sequence in , to obtain roots in the essential range of that belong to a lower-rank subspace of the span of . More precisely, given such that has nontrivial component on some , we will perform admissible perturbations on to obtain a new sequence with . Such a manipulation is possible only under some circumstances, which are enunciated in Propositions 5.5 and 5.6 below. First, the following proposition holds the basic Lie-algebraic calculations that make our “sliding along ” procedure work.
Proposition 5.3**.**
Assume that . Given a sequence in with unbounded, there exists in such that
- (1)
* is unbounded* 2. (2)
* is unbounded*
**Proof: ** The bilinear map induced by the bracket is nondegenerate; we recall the proof of this fact for real semisimple Lie algebras. Denote the Killing form on ; the Cartan involution as in section 2.5.1; and the dual with respect to of . Then, given nonzero, . Rescaling if necessary, the vectors , and form an -triple. Consider , which is an -module. If were zero, then would be a submodule with highest weight , which implies . On the other hand, is also an -module with lowest weight , which is impossible.
Given , (for any norm on ), let
[TABLE]
Then . In particular, there exist , such that
[TABLE]
is unbounded. Observe that replacing by gives the same conclusion with the extra property that . Now (1) is proved.
The conjugates in (2) are given, for some , by
[TABLE]
After replacing with , the components are
[TABLE]
From (1), the components of the terms corresponding to form an unbounded sequence. The following lemma shows that replacing by , with a suitable , makes the components unbounded too.
Lemma 5.4**.**
Let be sequences in a finite dimensional vector space . Assume that one of the sequences is unbounded. Then for a suitable choice of , the sequence is unbounded.
**Proof: ** There exist values of in , say , such that the vectors form a basis of . Let be any norm on . Then on the vector space of linear maps , we have two norms:
[TABLE]
and
[TABLE]
If denotes the linear map , then is unbounded (which is the case under the hypothesis of the lemma) if and only if is unbounded. The lemma follows.
Define to be the subset comprising the positive roots in which all occur with a positive coefficient. Observe that this set is nonempty only when is simple.
Proposition 5.5**.**
*(Transverse sliding)
Let with , and assume and all its admissible perturbations act equicontinuously with respect to segments on . Let such that for all , for all , if is a root, then it belongs to . Suppose for some . Then vertical and transverse perturbation of yields such that .*
**Proof: ** If is unbounded, there is nothing to do. By Proposition 5.1, we may assume after a vertical perturbation that is trivial for all for all , in particular for . Let for in . Then, for some ,
[TABLE]
By our hypotheses, , hence and . By Proposition 5.3, we can choose in such that the sequence is unbounded.
We have the relation
[TABLE]
We wish to show that . The action of on is scalar multiplication by , where , so it is enough to show that , for some constant . If this were not the case, then, up to taking a subsequence, there would be in with . For the product
[TABLE]
we know from above that . Thus , while , which contradicts the fact that acts equicontinuously with respect to segments.
Now let , which tends to 0. It is easy to verify that
[TABLE]
Thus is a transverse perturbation of according to Definition 2.8, and, because is unbounded, it has , as desired.
Proposition 5.6**.**
*(Vertical sliding)
Let and . Let with (). If (or , resp.), then left and right vertical perturbation of yields such that .*
**Proof: ** We can assume after vertical perturbation that . We apply proposition 5.3 to obtain in such that is unbounded, where
[TABLE]
for some , with . In this case, and together imply that for all . Thus .
Let . The lower bound on implies , so
[TABLE]
is obtained by left and right vertical perturbation from .
The proof for and unbounded is similar.
5.3. Algebraic proposition to reduce rank
Using the tools developed so far in this section, we will now state the algebraic proposition that drives our induction step. The next section contains the geometric interpretation of this result, and explains how to prove Theorem 3.1 by induction on .
Proposition 5.7**.**
Let be a sequence of with unbounded. Assume that , together with all its admissible perturbations, acts equicontinuously with respect to segments. Then an admissible perturbation of yields such that contains a root in .
The proof of this proposition is given in Sections 6.3 and 6.4 below.
6. Proof of Theorem 3.1 by induction on rank
The first half of this section gives the proof of Theorem 3.1 from Proposition 5.7. The second half gives the proof of Proposition 5.7.
6.1. Invariant parabolic subvarieties
Let with semisimple of real-rank and a parabolic subgroup with a Lie algebra , . Let be a parabolic subvariety through the base point . (These will be defined precisely below.) If acts equicontinuously with respect to segments on and preserves , then clearly it is equicontinuous with respect to segments on . The strategy for our induction argument is to find -invariant of rank less than .
Recall the notation introduced in Section 2.5.1, and denote by the Killing form on . Given a subset , let and be the ideals of and , respectively, commuting with . Let and , where the orthogonal is taken with respect to the scalar product . We obtain a subalgebra of
[TABLE]
It is easy to check that is -invariant, hence reductive, and has trivial center. It follows that is semisimple.
The corresponding connected subgroup is closed. Indeed, is a semisimple subalgebra of , hence is an algebraic subalgebra (see [6, Th 3.2]). For the corresponding Zariski closed subgroup of , the group is closed in , and so is its identity component .
A minimal parabolic of is contained in . The stabilizer of in contains and is algebraic, hence is a parabolic subgroup of , denoted . The orbit is a parabolic subvariety , nontrivial provided , and of rank less than .
Proposition 6.1**.**
Let and let be the exponential coordinates of . Then for any , the variety is invariant by . If for all , then acts trivially on ; if for all , then is trivial on .
**Proof: ** Let and .
Given as in the hypotheses above, , for all . Thus and for all . Thus , and acts trivially on .
Now let with for all . Write
[TABLE]
Note that unless , with a sum with negative integral coefficients of elements of and in ; in particular, has positive coefficient on some simple root of . In this case, is a positive root, so and . Thus , and is trivial on .
The above calculation with shows that is invariant by ; it is easy to see that leaves invariant. For invariance under a general sequence in , we can use the following basic lemma, the proof of which we leave to the reader:
Lemma 6.2**.**
Let be a simply connected nilpotent Lie group with Lie algebra . Let be an ideal of , and let be elements of and . Then there exists such that
[TABLE]
This lemma allows to write with and . We can then conclude because each factor , and preserves .
The unipotent radical of is with Lie algebra
[TABLE]
The analogue of in is . Note that
[TABLE]
and that the second factor is normal in . We will also need below the decomposition .
6.2. The induction step
Suppose that Theorem 3.1 holds for all parabolic models of real-rank at most . We will prove using Proposition 5.7 that it holds for all models of real-rank . Let of rank be given, and let be a sequence of which, together with all its admissible perturbations, acts equicontinuously with respect to segments. The aim is to show that is bounded. If not, then Proposition 5.7 gives, after an admissible perturbation, with containing a root .
There is a proper subset of such that . It cannot be that is contained in because . Now preserves by Proposition 6.1; denote the restriction by , which is a sequence of , and let be the decomposition into components on and , respectively. Because , it follows that is unbounded.
As , the induction hypothesis yields a contradiction, provided that all admissible perturbations of in act equicontinuously with respect to segments on . Admissible perturbation in means more precisely that vertical and transverse perturbations are as in Section 2.4 with in place of , and in place of , and Weyl reflections are done with respect to roots in . The following lemma ensures that satisfies the hypotheses of Theorem 3.1 and allows us to apply our induction hypothesis:
Lemma 6.3**.**
Let be a parabolic variety, and be a sequence of . Assume that preserves a parabolic subvariety on which it restricts to . If every admissible perturbation of acts equicontinuously with respect to segments in , then every admissible perturbation of in acts equicontinuously with respect to segments in .
**Proof: ** We will prove that any admissible perturbation of the sequence in can be obtained by an admissible perturbation of , restricted to . Assume that is obtained from by an admissible perturbation in . We seek an admissible perturbation of , such that preserves , and the restriction of to is precisely . Existence of such can be checked for each of the three kinds of admissible perturbations in :
(1) vertical perturbation: There are bounded sequences and in such that on . Because , the desired vertical perturbation of in is simply .
(2) transverse perturbation: In this case, write where and are two sequences of tending to [math]. As these are also sequences of , we can set ; we will show that this is a transverse perturbation in .
Let . Observe that because ,
[TABLE]
thus preserves and acts on it by . Taking gives , because the latter is in for all . This proves for all , and is a transverse perturbation of .
(3) Weyl reflection: Let realize the Weyl reflection , for . Decompose, using Lemma 6.2,
[TABLE]
where , , , and . By Proposition 6.1, both and are in the kernel of the restriction to , so we can write .
Now let be an automorphism of effecting on . Because , the derivative of preserves the Lie algebras , , and , so preserves the corresponding connected subgroups in . Thus , and
[TABLE]
preserves and restricts on it to , as desired.
The proof by induction of Theorem 3.1 is now complete, once we prove Proposition 5.7.
6.3. Proof of Proposition 5.7 (assuming the root system of is not of type )
Proposition 5.7 is vacuously true if the set is empty. Thus, we assume from now on that is a simple Lie group.
Let be a sequence of with unbounded. That means is nonempty. If it contains a root not in , then there is nothing to show, so we suppose that . Define the degree of to be the sum of the coefficients in the unique expression of as a positive integral linear combination of roots in .
Let . By Proposition 5.1, we may assume for . To prove that an admissible perturbation of results in with not contained in , we will show that for any of minimal degree, there is a sequence of admissible operations resulting in with the degree of strictly lower than the degree of .
Let of minimal degree. There is some with ; otherwise, would be in the negative of the Weyl chamber spanned by , contradicting that it is a positive root. For such ,
[TABLE]
Case . In this case, the Weyl reflection yields
[TABLE]
of smaller degree. The admissible operation yields with .
Case . Note that , because , and strings are unbroken.
If is not a maximal parabolic with , then , , and satisfy the hypotheses of Proposition 5.5, which thus gives another holonomy sequence with , which has lower degree than .
Now suppose is a maximal parabolic, with . Every root in has the form , where , and is in the positive integral span of . If none of the is a root, then again the hypotheses of Proposition 5.5 are satisfied, so, as above, there is a holonomy sequence with .
Thus we may assume that is a root for some .
Lemma 6.4**.**
Let be a maximal parabolic with . If for and , then is a valence-one vertex of the Dynkin graph of —that is, for exactly one element .
**Proof: ** The positive root can be written , with and for all . This statement is established by induction on the degree of together with the fact, seen above, that there exists at least one with . On the other hand, the expression in terms of simple roots is unique, and , so and for all . Denote this latter root .
Next we show by induction that the collection of roots corresponds to a connected set of vertices in the Dynkin diagram of for all . Assume connectedness for a given . If , then the -string of is symmetric, of the form . Note that because . Then is also a positive root, with . In other words, already belongs to the collection of roots appearing in , which is connected by the induction hypothesis.
We conclude that the elements of appearing in the decomposition of correspond to a connected subset of the Dynkin graph. These are precisely the elements of . As the Dynkin graph is a connected tree, the conclusion follows.
Let with . Write where , and let be the coefficient of in . The product
[TABLE]
(Although our root system is not necessarily reduced, the value 4 could only occur for or , neither of which is the case.) First suppose the Dynkin diagram has no double or triple edges, so the root system of is or . Then all roots of have the same length and . In this case, , so and . The -string of comprises and . Hence and . The -string of comprises . Now , so the Weyl reflection is an admissible perturbation resulting in with .
Under the assumption that is not of type , there are no triple bonds in the Dynkin diagram of , so it remains to consider the root systems with double bonds: , and . Let with as above be of minimal degree in Write , where —not necessarily a root—is a positive integral combination of elements of , and let be the coefficient of in . Because is a positive root,
[TABLE]
Write , numbered from left to right in the Dynkin diagram, where we follow the ordering of [7]. We have or .
Type or . For , the set comprises, for ,
[TABLE]
If is the short root, , then . The possibility is incompatible with (6). If , then the same inequality implies , so . If , then has lower degree, so a Weyl reflection is an admissible perturbation with the desired effect. Otherwise, and . In this case, as is an element of of minimal degree, . There is a rank-one subvariety left invariant by and on which it restricts to with unbounded. Proposition 4.1 leads to a contradiction.
If is the long root , then and , so (6) implies . Then or . In the first case, is a short root with . Proposition 5.6 permits vertical sliding along , resulting in with , or along , resulting in with . In the latter case, the Weyl reflection , so leads to the desired conclusion. Otherwise, for ; in this case, Weyl reflection in results in with a minimal element of of lower degree.
In , the set comprises from above, together with . If , then ; in this case, restricting to the rank-one subvariety yields a contradiction to Proposition 4.1.
Type . The set comprises, for ,
[TABLE]
If equals the long root, , then and . The inequality (6) gives and . If , then , and is a root of lower degree. The remaining possibility is with or simply . In the first case, the Weyl reflection results in with . In the second case, we again apply Proposition 4.1.
When equals the short root , then we first consider for . The Weyl reflection has lower degree. If , then , so Proposition 4.1 completes the proof.
Type . The roots in , in terms of the basis , are
[TABLE]
Recall that contains with a root in . The roots of correspond to those of when equals the long root and when equals the short root . In the first case the possibilities are
[TABLE]
The maximum degree in is 6. As all other roots of have degree at least 6, we may assume of minimal degree belongs to . If , then we can invoke Proposition 4.1. Otherwise, a Weyl reflection in or reduces the degree of and yields a new holonomy sequence with an element of lower degree in .
In the second case, contains the roots listed above, together with
[TABLE]
Now the maximal degree in is 7, and all other roots of have degree at least 7, so we may again assume . A Weyl reflection in or will reduce the degree of any , giving the desired conclusion in this case.
6.4. Proof of Proposition 5.7 for
Assume is of type , and write with . Then
[TABLE]
Assume first that , so . Given of minimal degree, the goal is to find an admissible perturbation with . As in the previous section (but with the roles of and switched), we can assume that . The two possibilities for are thus or . In the first case, is the only element of , so we can conclude using Proposition 4.1 as in the cases of and . In the second case, we apply Proposition 5.6. We can assume, after passing to a subsequence, that is bounded either below or above. If it is bounded below, then a vertical sliding on yields with , as desired. If is bounded above, then vertical slidings give in . Then the Weyl reflection on gives with .
Now consider , so . The condition leaves the possibilities or for . Unfortunately, the tools used above don’t help in either of these cases. The solution is to slide along , although it does not satisfy the hypotheses of Proposition 5.5.
Let be the reductive complement in a Levi decomposition of , where is simple of rank one. The group admits a decomposition, where as defined above, and is a maximal compact subgroup of . Write for the unipotent radical of . The decomposition of the corresponding Lie algebra into irreducible subspaces under is , where ; ; and . This decomposition can be seen from the fact that is contained in the sum of root spaces .
Recall that with if . Let in and , and set
[TABLE]
Just as in the proof of Proposition 5.5, and is a transverse perturbation of ; it is in particular a sequence in , although it may not be in . More precisely, , which can be deduced from the formula,
[TABLE]
with . Using Lemma 6.2, write with and . Proposition 5.3 gives that . Performing this transverse sliding twice if necessary, depending on , we arrive at .
Next, let be the decomposition of in . Finally, set
[TABLE]
Note that and , so . Clearly is a vertical perturbation of , so it is an admissible perturbation of . The conjugation by on preserves the subspace , so contains or . If it only contains , then we perform a Weyl reflection to finally obtain an admissible perturbation of with .
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