–buildings and base change functors
We prove an analog of the base change functor of –trees in the setting of generalized affine buildings. The proof is mainly based on local and global combinatorics of the associated spherical buildings. As an application we obtain that the class of generalized affine buildings is closed under taking ultracones and asymptotic cones. Other applications involve a complex of groups decompositions and fixed point theorems for certain classes of generalized affine buildings.
The so-called –trees have been studied by Alperin and Bass [AlperinBass], Morgan and Shalen [MorganShalen] and others and have proven to be a useful tool in understanding properties of groups acting nicely on such spaces. –trees are a natural generalization of –trees. Here is an arbitrary ordered abelian group replacing the copies of the real line in the concept of an –tree or the geometric realizations of simplicial trees.
Since simplicial trees are precisely the one-dimensional examples of affine buildings and real trees the one-dimensional –buildings, it was natural to ask whether there is a higher dimensional object generalizing –trees and affine buildings at the same time.
These objects, the so-called –affine buildings or generalized affine buildings, where introduced by Curtis Bennett [Bennett] and recently studied by the first author in [PetraThesis] and [Convexity2]. A recent application of them is a short proof of the Margulis conjecture by Kramer and Tent [KramerTent].
In the present paper, we address a generalization of an important geometric property of –trees: the existence of a base change functor. Easy to prove in the tree case, see for example [Chiswell], the generalization to –affine buildings turns out to be much harder.
We will prove that a morphism of ordered abelian groups gives rise to a base change functor mapping a generalized affine building defined over to another building which is defined over . In case is an epimorphism, we will see, that the pre-image under of a point in is again a generalized affine building, defined over the kernel of .
After having established our main results, we will present several applications. One of the consequences of our base change theorem is the proof of the fact that the class of generalized affine buildings is closed under taking asymptotic cones and ultracones.
1.1 Our main results
A set together with a collection of charts from a model space into is a generalized affine building if certain compatibility and richness conditions, as stated in Definition 2.1, are satisfied. These conditions imply the existence of a spherical building at infinity and of spherical buildings , called residues, around each point , see Section 2.2. The model space is defined with respect to a spherical root system and a totally ordered abelian group . Therefore it is sometimes denoted by . As a set it is isomorphic to the space of formal sums where is a basis of .
The model space carries an action of an affine Weyl group and the transition maps of charts are given by elements of . One of the conditions has to satisfy is, that every pair of points and is contained in a common apartment with .
Given a morphism of ordered abelian –modules and and let be an affine building with model space (or shortly ) and distance function which is induced by the standard distance on the model space. Then we have the following:
There exists a –building and (functorial) map such that maps to and such that
for all , where , are the distance functions defined respectively on and . Furthermore, the spherical buildings and at infinity are isomorphic.
The map will be referred to as the base change functor associated to .
Every morphism of abelian groups can be written as the composition of an epimorphism followed by a monomorphism. The kernel of an epimorphism of ordered abelian groups is convex in the sense that given some in this kernel then implies that is also contained in the kernel. Therefore the ordering of an abelian group induces an order on the quotient of by the kernel of an epimorphism. Hence morphisms of ordered abelian groups can also be decomposed into an epimorphism followed by a monomorphism.
We will prove the two cases, of an epimorphism and a monomorphism, separately in the first and third Main Result. Theorem 1.1 is a direct consequence of the combination of both.
In case is an epimorphism one defines the base change functor and the building is as follows. Let two points be equivalent, denoted by , when and let be the quotient of defined by this equivalence relation. Defining a metric on by the quotient map turns out to satisfy the properties needed for the first Main Result.
It is possible to define a set of charts from into such that the following theorem holds. For details see Section 3.
Main Result 1.
Let be an epimorphism of ordered abelian groups and a –building. Then the following hold:
There exists a –building and a map , called the base change functor associated to , such that the apartment system is mapped onto by and such that satisfies
for all , where , are the distance functions defined respectively on and . Moreover, the spherical buildings and at infinity are isomorphic.
These base change functors act as a functor on the category of –buildings to the one of the –buildings. In particular, let be a group acting on by isometries, then also acts on by isometries and the map is –equivariant.
As mentioned above, the kernel of an ordered abelian group is again such. It turns out that one can prove that the fibers of the base change functor are again generalized affine buildings. Details of the proof can be found in Section LABEL:section:secondMain.
Main Result 2.
Let be a base change functor associated to an ordered abelian group epimorphism , applied to a –building . For all elements of the following is true.
The set admits a set of charts making it into a –building with as distance function the distance function inherited from .
There is a natural isomorphism between and , where is as in the first Main Result.
One can prove a result similar to the first main one for monomorphisms of ordered abelian groups. The construction of the building is again of explicit nature. The basic idea is to take the product of the old charts with the new, enlarged, model space and consider equivalence classes of these products. Since this construction is more involved than the one in the first Main Result, let us postpone details to Section LABEL:section:thirdMain. There we will prove
Main Result 3.
Let be a monomorphism of ordered abelian groups and a –building. Then the following assertions hold
There exists a –building and a map satisfying
for all , where , are the distance functions defined respectively on and . Further maps the apartment system to and the spherical buildings and at infinity are isomorphic.
These base change functors act as a functor on the category of –buildings to the one of the –buildings, but only for isometries mapping apartments to apartments. In particular, let be a group acting on by isometries stabilizing the system of apartments, then acts on by isometries stabilizing the system of apartments and the map is –equivariant.
The proofs of the Main Results 1 to 3 can be found in Sections 3 to LABEL:section:thirdMain. Before defining generalized affine buildings in Section 2, we will use the penultimate subsection of the introduction to state our main applications. The last subsection will be devoted to an example clarifying the main results.
Let us quickly summarize the applications proved in Section LABEL:sec:applications.
Asymptotic cones of metric spaces capture the ‘large scale structure’ of the underlying space. The main idea goes back to the notion of convergence of metric spaces by Gromov in the early 80’s (see [Gromov2]) and was later generalized using ultrafilters by van den Dries and Wilkie [vandenDriesWilkie]. Asymptotic cones provide interesting examples of metric spaces and have proven useful in the context of geometric group theory.
In Section LABEL:sec:cones we will prove, using the base change functor, that the class of generalized affine buildings is closed under ultraproducts, asymptotic cones and ultracones. The main results read as follows:
Theorems LABEL:prop:ultraproducts and LABEL:thm:cones.
The ultraproduct of a sequence of –affine buildings defined over the same root system is again a generalized affine building over .
Asymptotic cones and ultracones of generalized affine buildings are again such.
Furthermore, if is modeled on , then its asymptotic cone is modeled an and its ultracone on .
In particular we have shown that asymptotic cones of –buildings are again such, and with this yielding an alternative proof of the same result shown earlier with completely different methods by Kleiner and Leeb (see [KleinerLeeb]).
Fixed point theorems
The base change functors can be used to reduce problems of generalized affine buildings to the (easier) case of –buildings. In Section LABEL:section:reducing we illustrate this with a fixed point theorem for a certain class of –buildings (we postpone the description of this class to the aforementioned section). The result then reads:
A finite group of isometries of a generalized affine building of this class admits a fixed point.
Complex of groups decompositions
One can use the first Main Result to conclude that groups acting nicely on certain affine buildings do admit a complex of groups decomposition. We will not carry out the details of the proof, but let us make the statement a bit more precise.
Assume that is modeled over an abelian group , where the two components are ordered lexicographically, and assume further that the image of the base change functor associated to the projection is a simplicial affine building.
Then, if is a subgroup of the automorphism group of such that the induced action on is simplicial, the group has a complex of groups decomposition where each vertex group acts on a –building. In addition, if the action of on is free, then each vertex group acts freely on a –building.
1.3 An example
Let us illustrate the main results with an example in an algebraic setting. We start with describing a class of generalized affine buildings (following [BennettThesis, p. 97]). Let be a field with a valuation to an ordered abelian group . Then one can define root group data with valuation for the special linear group (). These data give rise to an -dimensional –building admitting a natural action of . The spherical building at infinity here is the spherical Tits building associated to . The thick residues are isomorphic to the spherical Tits building associated to , where is the residue field of the pair .
Let be the lexicographically ordered group . Let be some field with a –valued valuation . An example could be a rational function field in the variable allowing powers in . As mentioned above one can associate a –building to the group .
Let be the group ordered lexicographically. Let be the map
This is a morphism of ordered abelian groups which can be split up in an epimorphism and monomorphism (so ) where
The image of under the base change functor for is the generalized affine building for , but now using the (real) valuation . Similarly the image under the base change functor for will be the generalized affine building associated to with the valuation with values in . Theorem 1.1 mentions that the spherical buildings at infinity of these generalized affine buildings are isomorphic, this is reflected in this example by the special linear group staying the same.
To illustrate the second Main Result consider a point with a thick residue in the generalized affine building associated to with the valuation . This residue is isomorphic to the spherical Tits building for . The second Main Result now states that the preimage of this point is an –building with as building at infinity this residue. This –building is the one defined by the (real) valuation on the residue field induced by the valuation .
In this section we will define –buildings and state some basic results about them for use in later sections. For a detailed study of generalized affine buildings and proofs of the results in this introductory section we refer to [Bennett] and [PetraThesis].
2.1 Definition of apartments and buildings
We will first define the model space for apartments in –buildings and examine its metric structure. We conclude this subsection with the definition of a –building.
For a (not necessarily crystallographic) spherical root system let be a subfield of the reals containing the set of all evaluations of co-roots on roots. Notice that can always be chosen to be the quotient field of . If is crystallographic this is . Assume that is a (non-trivial) totally ordered abelian group admitting an –module structure and define the model space of a generalized affine building of type to be the set
We will often abbreviate by . A fixed basis of the root system provides natural coordinates for the model space . The vector space of formal sums
is canonically isomorphic to . The evaluation of co-roots on roots is linearly extended to elements of . Let be the point of corresponding to the zero vector.
By a set of positive roots is defined which determines the fundamental Weyl chamber
with respect to . By replacing some (which might be all or none) of the inequalities in the definition of by equalities we obtain faces of the fundamental Weyl chamber.
The spherical Weyl group of acts by reflections , , on the model space . The fixed point sets of the are called hyperplanes and are denoted by . One has .
An affine Weyl group is the semidirect product of a group of translations of by . If equals , then is called the full affine Weyl group. The actions of and on induce an action of on . An (affine) reflection is an element of which is conjugate in to a reflection , . A hyperplane in is the fixed point set of an affine reflection . It determines two half-spaces of called half-apartments.
We define a Weyl chamber in to be an image of a fundamental Weyl chamber under the affine Weyl group . The image of the faces of the fundamental Weyl chamber then define the faces of this Weyl chamber. A face of a Weyl chamber will also be called a Weyl simplex. Note that a Weyl simplex contains exactly one point which is the intersection of all bounding hyperplanes of . We call it the base point of and say is based at .
Let be a totally ordered abelian group and let be a set. A metric on with values in , short a –valued metric, is a map satisfying the usual axioms of a metric. That is positivity, symmetry (), equality if and only if and the triangle inequality for arbitrary triples of points. The pair is called –metric space.
A particular –invariant –valued metric on the model space is defined by
A subset of is called convex if it is the intersection of finitely many half-apartments. This includes the empty set and . The convex hull of a subset is the intersection of all half-apartments containing .
Note that Weyl simplices and hyperplanes, as well as finite intersections of convex sets are convex. A convex hull of a subset of the model space is not necessarily convex due to the finiteness requirement.
Let be a set and a collection of injective maps , called charts. The images of charts are called apartments of . Define Weyl chambers, hyperplanes, half-apartments, … of to be images of such in under any . The set is a (generalized) affine building with atlas (or apartment system) if the following conditions are satisfied
If and then .
Let be two charts. Then is a convex subset of . There exists with .
For any two points in there is an apartment containing both.
If and are two Weyl chambers in there exist sub-Weyl chambers in and an such that .
For any apartment and all there exists a retraction such that does not increase distances and .
If and are charts such that the associated apartments intersect pairwise in half-apartments then .
The dimension of the building is , where .
Conditions (A1)-(A3) imply the existence of a –distance on , that is a function satisfying all conditions of the definition of a –metric but the triangle inequality. Given in fix an apartment containing and with chart and let in be defined by . The distance between and in is given by . By Condition (A2) this is a well-defined function on . Therefore it makes sense to talk about a distance non-increasing function in (A5). Note further that, by (A5), the defined distance function satisfies the triangle inequality. Hence is a metric on .
In the case of –buildings one has that Condition (A6) follows from the other conditions. This can be found (along with other equivalent definitions for this particular case) in [Parreau]. One can also define –buildings in a more geometric way, see [KleinerLeeb]. There is a paper in preparation by the first author investigating alternative definitions for generalized affine buildings.
2.2 Local and global structure of –affine buildings
There are two types of spherical buildings associated to an affine –building of type : the spherical building at infinity and at each point a so-called residue .
Two subsets of a –metric space are parallel if there exists such that for all there exists an such that for . Note that parallelism is an equivalence relation. One can prove
[Bennett, Section 2.4] Let be the model space equipped with the full affine Weyl group . Then the following is true.
Two hyperplanes (or two Weyl simplices) are parallel if and only if they are translates of each other by elements of .
For any two parallel Weyl chambers and there exists a Weyl chamber contained in and parallel to both.
A simplex in the spherical building at infinity is a parallel class of a Weyl simplex in . Hence as a set of simplices
One simplex is contained in a simplex if there exist representatives which are contained in a common apartment with chart in , having the same base point and such that is contained in .
The set defined above is a spherical building of type with apartments in a one-to-one correspondence with apartments of .
See [BennettThesis, 3.6] or [PetraThesis, 5.7].
To define a second type of equivalence relation on Weyl simplices we say that two of them, and , share the same germ if both are based at the same point and if is a neighborhood of in and in . It is easy to see that this is an equivalence relation on the set of Weyl simplices based at a given point. The equivalence class of , based at , is denoted by and is called germ of at . The germs of Weyl simplices based at a point are partially ordered by inclusion: if there exist representatives contained in a common apartment such that is a face of . Let be the set of all germs of Weyl simplices based at . Then
[PetraThesis, 5.17] For all the set is a spherical building of type which is independent of .
Let be a germ of a Weyl simplex based at . We say that is contained in a set if there exists an in such that is contained in .
The following properties will be of use in subsequent proofs of the present paper.
Let be an affine building of type . Then:
Let and be two Weyl chambers based at the same point . If their germs are opposite in then there exists a unique apartment containing and .
For any germ the affine building is, as a set, the union of all apartments containing .
See 5.23 and 5.13 of [PetraThesis] for a proof.
The proof of the following proposition is the same as of Proposition 1.8 in [Parreau]. A consequence of it is that given a point in and parallel class of Weyl simplices, there is a unique Weyl simplex in this class based at the given point.
Let be an affine building and a chamber in . For a Weyl chamber based at a point there exists an apartment with chart in containing a germ of at and such that is contained in the boundary .
Given a germ of a Weyl chamber in a fixed apartment one can define a retraction of the building onto as follows.
Fix a germ of a Weyl chamber in . Given a point in let be a chart in such that and are contained in . Define
where is such that . The map is called the retraction onto centered at .
By Condition (A2) and item 2 of Proposition 2.5 this retraction is well-defined. Furthermore, as proved in Appendix C of [PetraThesis], it is distance non-increasing. Furthermore, the restriction of to an apartment containing is an isomorphism onto .
We end these preliminaries by pointing out that our main results (in particular Main Result 2, part 2) allow for more spherical buildings to be defined from a –building than the two constructions mentioned in this section. In fact, one can associate a spherical building to each set of points with distance in a convex subgroup of from a certain point of . The spherical building at infinity and the residues correspond to the choices and .
3 Proof of the first Main Result
Given an epimorphism of ordered abelian –modules and , we define the base change functor as follows. Let be an affine building with model space (or shortly ) and distance function which is induced by the standard distance on the model space.
The relation “” on with when is an equivalence relation (due to the triangle inequality). Let be the quotient of defined by this equivalence relation. The associated quotient map is surjective by definition. One can define a metric on by putting . This metric is well-defined due to the triangle inequality, one also easily checks it is indeed a metric. Let be the model space and the associated affine Weyl group. In the same way as for one can define a map from the model space to . For each chart one has that the preimages of the maps and on are the same. Hence one can define an injective map such that equals .
This way we have defined a set of charts from into . Automatically we also have defined (half-)apartments, hyperplanes, Weyl chambers, … in . By construction these objects are the images under of similar objects in .
Conditions (A1) and (A3)-(A5) for are easy consequences of the fact that these conditions are already satisfied by . The only non-trivial condition to check is Condition (A2). This turns out to be particularly difficult when two non-intersecting apartments intersect after applying . Condition (A6) follows as a byproduct of the proof of (A2).
The outline of the proof is the following. We start with investigating images of pairs of already intersecting convex sets (Lemma LABEL:lemma:compare). This will imply that for two already intersecting apartments nothing surprising happens (Lemma LABEL:cor:A2part).
The next step is then to investigate local structures, i.e. the residues. The easier case of already intersecting apartments will be sufficient to show that these local structures are spherical buildings (Lemma LABEL:lemma:spheric). Condition (A6) follows from this case as well. The results we obtain in this part are also useful to prove the second Main Result later on.
This local information eventually allows us to prove (A2) in full generality (Lemma LABEL:lemma:a2general). After this we end by showing functoriality.
3.1 Intersecting convex sets
In this section we study how already intersecting convex sets behave under the map . These lemmas will be used later on to investigate the local structure of the quotient space and in the proof of Condition (A2).
Let and be two points of the model space lying in respectively two Weyl simplices and both based at some point . Suppose that . Then if and do not have Weyl simplices in common, other than the base point, one has that .
The images of the two Weyl simplices and under are again two Weyl simplices and having no common Weyl simplices. So the intersection of and is the singleton . As the point lies in this intersection, one has that . By the definition of , we conclude that .
Suppose that some subset of the model space is closed under taking convex hulls of pairs of points of . Then a germ based at a point lies in , if and only if, there is a point contained in the Weyl simplex in corresponding to that germ, and is the minimal Weyl simplex containing .