A structure theorem for generalized affine buildings
We prove that a generalized affine building has a weak spherical building at infinity if and only if it splits as a product of a euclidean space and a generalized affine building with a thick spherical building at infinity. This can be seen as canonical extension of the structure theorem for weak spherical buildings. The class of generalized affine buildings under considerations includes (but is larger than) the class of euclidean -buildings.
In his paper on finite reflection groups that appear as Weyl groups [Tit77] Tits showed, that a thick spherical building never is of type or of any type that has as a standard parabolic.
The natural question then was how the weak, i.e. non-thick, spherical buildings of these types do arise. In the same paper Tits gave a construction for weak buildings of type and based on sub-diagrams of their diagram.
Later Scharlau [Sch87] found a systematic way to describe all weak spherical buildings. The main idea was already proposed by Tits: given a weak building one considers equivalence classes of chambers with respect to thin panels. These thin-classes then form the chambers of a thick spherical building, called the thick frame of the weak building.
Scharlau’s paper is complemented by Sarah Rees thesis [Ree88] which shows how to interpret the statement in the realm of incidence geometries. Caprace later showed in [Cap05] that Scharlau’s result also extends to twin buildings, which are associated with Kac-Moody groups.
One can easily see that such a structure theorem will not hold within the class of simplicial affine buildings. One of the reasons simply is, that some of the spherical Coxeter groups involved do not have a discrete affine counterpart. One may, however, always consider a non-discrete affine Weyl group which is simply a semidirect product of a spherical Coxeter group with some -invariant translation group . Replicating the structure theorem to the spherical building at infinity we show that Scharlau’s construction canonically extends to a certain class of generalized affine buildings. The class of generalized affine buildings we consider is described in Section 3. It is worth mentioning that our construction also works for buildings whose metric structure is not euclidean when restricted to an apartment, but some Weyl compatible real-valued (asymmetric) metric. The precise statements of the main results are as follows.
Theorem A (Subdivided product structure).
Let be a generalized affine building with affine Weyl group and suppose that the spherical building of type at infinity is thick. Then for any spherical Coxeter group of rank , with , that contains the rank group as a reflection subgroup, there exists a generalized affine building such that is (as a simplicial building) isomorphic to the thick frame of the spherical building .
The building with atlas in the statement of the theorem above is constructed as follows. Define to be the set equipped with the set of charts with an affine Weyl group defined to be for some -invariant . We will prove in Section 4 that the pair is indeed a generalized affine building with affine Weyl group and that is (as a simplicial building) isomorphic to the thick frame of the spherical building . Moreover is independent of the choice of .
Conversely one can show the following splitting theorem.
Theorem B (Splitting theorem).
Let be a generalized affine building with affine Weyl group with spherical of rank and suppose the thick frame of is of type . Then splits as a product where is a generalized affine building with and .
If in addition is -invariant, then is a generalized affine building with affine Weyl group and can be constructed from as in creftypecap A.
One may call the soul (or again thick frame) of having the thick frame of as its boundary.
(enrichment of the translation sub-group) In general, unlike in the spherical case, the constructions in the two main theorems are not mutually inverse, that is . Applying the reverse construction in most cases substantially enlarges the translation part of the underlying Weyl group. This is in particular true even if .
Our proofs of creftypecap A and B, provided in Section 4, rely on new axiomatic characterizations of (generalized) affine buildings due to Bennet and Schwer [BSS14]. The class of generalized affine buildings and its axiomatic description is introduced in Section 3, where we also explain the construction of their spherical building at infinity. The main feature is the independence of the choice of some Weyl-compatible metric and the fact that the characterization does not involve retractions onto apartments. Both of these features are part of the original definition due to Tits [Tit86] as well as the equivalent characterizations due to Parreau [Par00].
It is natural to ask what the relation of the presented splitting theorem to the standard CAT(0) splitting of metric spaces is. This relation is discussed in Section 5. We proceed below by quickly discussing spherical buildings, their geometric realizations and Scharlau’s structure theorem in Section 2.
2. The structure theorem for spherical buildings
The main goal of this section is to state and explain Scharlau’s structure theorem for weak spherical buildings.
Structure of Coxeter groups
We assume that the reader is familiar with Coxeter systems and the structure of their Coxeter complexes . Recall that for a Coxeter matrix .
Definition 2.1 (action on a vector space).
The Coxeter group acts by linear transformations on a real vector spaces with basis and preserves the bilinear form defined by and if . The form is a scalar product if and only if is finite. In this case acts as a group of orthogonal transformations, and the action is completely defined by its restriction to the unit sphere. Hence is called a spherical Coxeter group.
From now on will always be spherical.
Definition 2.2 (cones and the Weyl complex).
The reflection hyperplanes of in subdivide into closed convex cones. The set of cones and their faces is ordered by inclusion and carries the structure of a simplicial complex isomorphic to . We call the Weyl complex of and refer to its elements as Weyl simplices. The intersection with the Weyl simplices induce a simplicial structure on the unit sphere in called the geometric realization of . Multiplication by fixes every reflection hyperplane and hence induces an (involutory) isomorphism of or , called the opposition involution.
We are now going to show how an inclusion of spherical Coxeter groups determines a geometric inclusion of their geometric realization.
Lemma 2.3 (geometric splitting of ).
Let be spherical, and a reflection subgroup of rank with generators corresponding to hyperplanes in . Then has dimension and splits as the orthogonal sum with for hyperplanes in .
Using this splitting lemma one can describe what happens on the level of the spherical Coxeter complex, respectively with its geometric realization.
Remark 2.4 (direct sums and spherical joins).
It is easy to see that . One can thus reconstruct the complex from by forming the direct sum and then subdivide by the walls of .
On the level of the unit sphere a direct sum with becomes a spherical join with in the sense of Definition 5.13 of [BH99]. In particular
We call the -fold suspension of . Be aware that the join does depend on the underlying isomorphism.
We take the chance to remind the reader that the geometric realization of the simplicial join of spherical Coxeter complexes is isometric to the spherical join of their geometric realizations and that the same is true for spherical buildings, see for example [CL01].
Remark 2.5 (inclusion of Coxeter groups).
An inclusion of Coxeter groups as a reflection subgroup is only determined up to conjugation, so the subdivision depends on the conjugacy class of such an embedding. Two examples of such embeddings are shown in Figure 1.
Let us move on by recalling the definition of a simplicial building.
Definition 2.6 (simplicial buildings).
Let be a simplicial complex and suppose that is the union of a collection of subcomplexes , called apartments, each of which is isomorphic to some Coxeter complex. The complex is a (simplicial) building if the following two axioms are satisfied.
For every pair of simplices there exists an apartment containing and .
If and are two apartments containing both and then there is an isomorphism fixing and pointwise.
A building is spherical, respectively affine, if the corresponding Coxeter group is spherical or affine.
It is easy to see that all apartments in a building must be of the same Coxeter type, which we call the type of the building. The maximal simplices in a building are all of the same dimension and will be called chambers, their codimension one faces are called panels.
Definition 2.7 (thickness).
A panel in a building is thick if it is contained in at least three chambers. A panel is thin if it is contained in exactly two chambers. A building is thick (or thin) if all its panels are. If a building contains at least one thin panel we call it a weak building.
It is clear from the definition of a building that panels are thin if and only if they are not thick. Buildings are weak if and only if they are not thick. The thin buildings are exactly the Coxeter complexes.
This subsection concerns spherical buildings only. We will now explain the main ingredients of Scharlau’s proof which are summarized in the following two Lemmata. They allow to distinguish between thick and thin walls in a weak building, and imply that there is a subgroup of the Weyl group generated by reflections in thick walls.
The following is Lemma 1 in [Sch87].
Lemma 2.8 (thick and thin walls).
Let be a spherical building and a wall in . Then the panels contained in are either all thick or all thin.
Fix an apartment that has as a wall and let be distinct panels in . Choose a minimal gallery of panels in . This is possible as walls are themselves chamber complexes as e.g. shown in Le. 4.1 of [Cap05]. Consider . Since both and are in they must be opposite in , which is a generalized polygon. A standard result of Tits [Tit74] Theorem 2.40 then implies that the stars along the gallery are all isomorphic, i.e.
Hence the lemma. ∎
Once we have identified thick and thin walls in a Coxeter complex we may look at the reflection subgroup of the Weyl group generated by the reflections on thick walls only. In fact the set of reflections in thick walls is as the set equal to the group they generate.
This follows from the next lemma, which is Lemma 2 of [Sch87].
Lemma 2.9 (thickness is reflection invariant).
Let be an apartment in a spherical building, let be a thick panel in , and the half-apartments determined by , and let be the folding onto . Then maps thick panels to thick panels and thin panels to thin panels.
We are now going to ignore the thin walls and “glue” chambers together to form “larger” chambers with only thick panels.
Definition 2.10 (thin-classes).
Two chambers and in a spherical building are thin-adjacent if is a thin panel. A gallery is called thin if and are thin-adjacent for any , and we say that and are thin-equivalent if they can be joined by a thin gallery. It is easy to see that thin-equivalence is in fact an equivalence relation on , and we refer to the corresponding equivalence classes as thin-classes. For a chamber we denote its thin-class by .
One immediately has the following properties.
Lemma 2.11 (properties of thin-classes).
For every (weak) spherical building the following hold.
If two chambers and do not belong to the same thin-class, then every gallery from to crosses a thick wall.
Thin-classes are convex.
If an apartment contains a chamber , then .
Geometrically the thin-classes are the (closures of the) connected components after having removed the thick walls. So for every apartment in a spherical building the group generated by reflections in thick walls is a Coxeter group (by creftypecap 2.9) which acts simply-transitively on the set of thin-classes. This defines an apartment by “forgetting” some of the structure in . Overall we obtain a building as the union of all apartments , where is an apartment in . We summarize this in the following Definition/Proposition.
Definition/Proposition 2.12 (thick frame).
For a (weak) spherical building the set of thin-classes in is the set of chambers of a chamber complex called the thick frame of . By construction is a union of apartments.
We are now ready to state Scharlau’s structure theorems as shown in [Sch87].
Theorem 2.13 (spherical splitting).
The thick frame of a spherical building of type is a thick spherical building of type . Two chambers of are adjacent if and only if they have adjacent representatives in . The apartments of are in one-to-one correspondence with the apartments of . The Coxeter group associated to an apartment in is the subgroup of generated by the reflections along thick walls of an apartment in , and this way the Weyl group becomes a subgroup of the Weyl group , which is determined up to conjugation.
Theorem 2.14 (subdivided suspension).
Conversely, if is a thick spherical building of rank and some spherical Coxeter group of rank containing as a reflection subgroup, then subdividing the -fold suspension by the walls of defines a weak building with thick frame . This subdivision only depends on the conjugacy class of the inclusion .
3. Generalized buildings
In this section we introduce the class of generalized affine buildings for which we will prove a natural extension of Scharlau’s construction.
We will only consider generalized affine buildings with apartments isomorphic to . The full generality in which the following definition makes sense is explained in [BSS14]. We start with a definition of generalized Weyl groups that will be considered on .
Definition 3.1 ((generalized) affine Weyl groups).
Let be a spherical Coxeter group, let be the affine space associated with the vector space from creftypecap 2.1, and a -invariant translation subgroup of the automorphism group of . Then is called an affine Weyl group and the images of Weyl simplices under are called (affine) Weyl-simplices. Affine Weyl chambers are sometimes called sectors.
An element is called an affine reflection if is a reflection in and is orthogonal to the corresponding hyperplane . This is equivalent to the fixed point set of being non-empty. The fixed point set of an affine reflection is called an affine hyperplane and splits into to half spaces, and we say or separates if and are contained in different open half spaces with respect to .
Affine Weyl-simplices are the translates of Weyl-simplices under and thus are closed, convex subsets of . If is of rank , then .
Definition 3.2 (Weyl compatible metric).
A metric on is called Weyl-compatible if it is -invariant satisfies the following conditions
For all Weyl chambers and Weyl simplices of the same type, if there exists such that for all sub-faces there exists a sub face such that for there exists with for all , then is a translate of .
If separates and , then there is a such that .
This ensures that two sector faces of the same type are at bounded distance of one another if and only if they are translates.
Since the euclidean metric on is always Weyl compatible as shown in [BSS14] Chapter 10, and we only consider generalized affine buildings with apartments isomorphic to , it is convenient to think of the topology induced by as the euclidean topology.
We follow [BSS14] and define a class of generalized affine buildings as spaces constructed from a model space which satisfy some extra conditions.
Definition 3.3 (spaces modeled on ).
Let and as in creftypecap 3.1. A space modeled on is a pair with any set together with an atlas , that is a family of injective charts , satisfying the following axioms:
For any and , .
For any with , the preimage is closed and convex in and for some .
For any pair of points there exists such that .
The sets , for are called apartments of .
In particular the axioms imply that is as a set the union of its apartments and transition maps are given by elements of . Therefore the notion of Weyl chambers and their germs makes sense for .
Definition 3.4 (germs of Weyl simplices).
Let be Weyl simplices in or in both based at some point .
We say that and share the same germ if is a neighborhood of in and .
This is an equivalence relation on the set of -based Weyl simplices, and we denote the equivalence class of , called the germ of in by . The set of all -based germs is ordered by inclusion.
In an apartment , the set of -based Weyl simplices is isomorphic to and two -based germs are opposite if they are contained in a common apartment and are images on one another under the opposition involution. For Weyl chambers this means they are images of one another under the (unique) longest element of the spherical Weyl group. Two Weyl chambers are opposite at , if their germs are opposite.
We may now define generalized affine buildings.
Definition 3.5 (generalized affine buildings).
A space modeled on is a generalized affine building if the following two axioms are satisfied.
Any two germs of Weyl chambers based at the same vertex are contained in a common apartment.
Any two opposite Weyl chambers based at the same vertex are contained in a unique apartment.
Remark 3.6 (geometric meaning of a germ).
One can think of the germs in as the set of local (initial) directions in which one go from . Then (GG) essentially says that any pair of possible directions is represented in some apartment. In fact the set of germs of Weyl chambers based at a point can be thought of as the set of chambers of a spherical building , the building of germs at . This is is the analogue of a star or link of a vertex in the simplicial setting.
Remark 3.7 (axiom (A4)).
Remark 3.8 (considered class of affine buildings).
The class of generalized affine buildings under consideration is much richer than the class of simplicial affine or even euclidean buildings (in the sense of [Par00]). It contains affine buildings, as well as euclidean buildings, leafless -trees, and flat cones over (the geometric realization of) spherical buildings. In fact creftypecap A says that any leafless -trees gives rise to generalized affine buildings of arbitrary rank. Moreover it does contain geometric realizations of affine buildings equipped with (asymmetric) metrics different from the euclidean metric.
Spherical buildings at infinity
We spend the rest of this section to explain that the structure at infinity of a generalized affine building is a spherical building in the sense of creftypecap 2.6. Proofs can be found in [BSS14] Chapter 9.
Definition 3.9 (parallel Weyl simplices).
Let be a generalized affine building. Two Weyl simplices and are parallel if there is a sequence such that and are translates of each other in some apartment for .
In the light of (A4) this means that two Weyl chambers are parallel if and only if they have common sub-Weyl chamber and that such a sequence can always be chosen with .
Note that Weyl simplices are defined with respect to a certain (fixed) atlas of , and so is parallelism. It is easy to see that parallelism is an equivalence relation and we denote the parallel-class of a Weyl simplex by or simply and the set of all parallel classes of Weyl simplices in by or .
Sometimes may carry more than one possible atlas, say , and in this case .
Remark 3.10 (parallelism in case of metric buildings).
In case the building is equipped with a Weyl compatible metric one can show that two Weyl simplices are parallel if and only if they have finite Hausdorff-distance with respect to .
Theorem 3.11 (the spherical building at infinity).
Let be a generalized affine building. The set of parallel classes of Weyl simplices of a generalized affine building of type is a spherical (simplicial) building of type . We call the spherical building at infinity of . The apartments of are in one-to-one correspondence with the apartments of .
Note that does depend on the choice of , but not on .
4. Proofs of creftypecap A and creftypecap B
This section contains the proofs of our main theorems. We will be working with generalized affine buildings whose apartments are copies of for some . However, we do not require them to be equipped with the euclidean metric.
We omit proofs for the part about , as they follow directly from creftypecap 3.11.
Proof of creftypecap A.
By construction, apartments in are of the form with the affine space associated with . Since the elements of correspond by construction to the elements in , (A1) and (A3) are satisfied.
Suppose . By pre-composing with suitable (and possibly different) elements of we can assume and thus splits as a product , with some non-empty and thus closed convex set. Hence and differ by some element of and thus (A2).
By construction a Weyl chamber in is a subset of where is a Weyl chambers in . Suppose and are opposite Weyl chambers based on the same vertex in , arising from Weyl chambers and in . We can assume that and are based on the same vertex, thus they determine a unique apartment in , and is the unique apartment in containing and , hence (CO).
Similarly if and are the germs of Weyl chambers and in , arising from Weyl chambers and in , we can assume they are based on the same vertex, say . Then and are contained in some apartment in , hence (GG). ∎
Proof of creftypecap B.
Let and be representatives of chambers and respectively. Then there is an apartment containing sub Weyl chambers and of and respectively, and by applying some translation we may without loss of generality assume they are based on the same vertex. In particular if and are opposite chambers in , they determine a unique apartment in , which proves the last part of creftypecap 3.11.
Similarly if two apartments and contain representatives and of a chamber , there is a common sub Weyl chamber , and we may without loss of generality assume that and agree on . If on the other hand and are apartments in , then the isomorphism in (B2) can by chosen to be type-preserving, i.e. an element of , and since translations are invisible to (or parallelism) such a splitting must occur apartment-wise.
5. A metric point of view
In this section we will discuss the connection with the CAT(0) splitting theorem. As before the generalized affine buildings have apartments that are copies of for some equipped with an arbitrary Weyl compatible metric defined in creftypecap 3.2. Any Weyl compatible metric on the model space or an apartment extends to a metric on , which we also denote by .
Example 5.1 (generalized buildings and the CAT(0) property).
Generalized affine buildings are in general not CAT(0) spaces. One can see that the CAT(0) property fails already on an apartment level. For example, the maximum norm induces a or - invariant (and compatible) metric on which is not uniquely geodesic and hence not CAT(0).
If is the euclidean metric, then Charney and Lytchak show in Proposition 2.3 of [CL01] that generalized affine buildings are CAT(0) spaces that have the geodesic extension property.
We will therefore discuss in this section the metric implications of our main theorem in connection with Bridson’s CAT(0)-splitting theorem, see Theorem 9.24. in [BH99].
Theorem 5.2 (CAT(0) splitting).
Let be a complete CAT(0) space in which all geodesic segments can be extended to (bi-infinite) geodesic lines. If is isometric to the spherical join of two non-empty spaces and , then splits as a product where for .
As pointed out in 6.1 of [CL01] thin panels of a spherical building are invisible to the metric. Hence a weak spherical building is indistinguishable from the -fold suspension of a thick spherical building. In the sense of creftypecap 2.13, the thin buildings come from some inclusion .
The CAT(0) splitting theorem readily allows us the identify as one of the splitting factors in creftypecap B, but a priori it is not clear that the second factor carries the structure of a generalized affine building, or even as a space modeled on . In order to establish such a characterization one necessarily needs at least to find a one-to-one correspondence of apartments as done in the proof of creftypecap B – at which point CAT(0) becomes superfluous as the actual metric in the space has no influence on the proof of creftypecap B.
However the CAT(0) case imposes further restriction on the metric. The metric necessarily is the product metric , making a convex subset of .
- [BH99] Martin R Bridson and André Haefliger. Metric spaces of non-positive curvature, volume 319. Springer Science & Business Media, 1999.
- [BSS14] Curtis D Bennett, Petra N Schwer, and Koen Struyve. On axiomatic definitions of non-discrete affine buildings. Advances in Geometry, 14(3):381–412, 2014.
- [Cap05] Pierre-Emmanuel Caprace. The thick frame of a weak twin building. Advances in Geometry, 5(1):119–136, 2005.
- [CL01] Ruth Charney and Alexander Lytchak. Metric characterizations of spherical and euclidean buildings. Geometry & Topology, 5(2):521–550, 2001.
- [Par00] Anne Parreau. Immeubles affines: construction par les normes. In Crystallographic Groups and Their Generalizations: Workshop, Katholieke Universiteit Leuven Campus Kortrijk, Belgium, May 26-28, 1999, volume 262, page 263. American Mathematical Soc., 2000.
- [Ree88] Sarah Rees. Weak buildings of spherical type. Geometriae Dedicata, 27(1):15–47, 1988.
- [Sch87] Rudolf Scharlau. A structure theorem for weak buildings of spherical type. Geometriae Dedicata, 24(1):77–84, 1987.
- [Tit74] Jacques Tits. Buildings of Spherical Type and Finite BN-Pairs, volume 386. Springer, 1974.
- [Tit77] Jacques Tits. Endliche Spiegelungsgruppen, die als Weylgruppen auftreten. Inventiones mathematicae, 43(3):283–295, 1977.
- [Tit86] Jacques Tits. Immeubles de Type Affine, pages 159–190. Springer Berlin Heidelberg, Berlin, Heidelberg, 1986.