Loop Groups and Twin Buildings

In these notes we describe some buildings related to complex Kac–Moody groups. First we describe the spherical building of SL_n() (i.e. the projective geometry PG(ⁿ)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field (z). Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac–Moody algebra. Next, we introduce topological twin buildings. These buildings can be used for a proof which is a variant of the proof by Quillen and Mitchell, of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac–Moody groups.Supported by a Heisenberg fellowship by the Deutsche Forschungsgemeinschaft.     

On Opposition in Spherical Buildings and Twin Buildings

In this paper, we prove a combinatorial property of twin apartments and opposition of chambers in twin buildings. We then characterize adjacency of chambers in twin buildings by meansof opposition of chambers. As an application, we study maps which satisfy certain conditions related to opposition of chambers, e.g., maps that preserve opposition. Applied to the special case of spherical buildings, all our main results as well as their corollaries are new.     

Affine Twin Buildings

It is well known from work of Bruhat and Tits that an affinebuilding has a spherical building at infinity. The paper studiesthe structure at infinity for an affine twin building. It isshown that the twinning restricts the structure at infinityin a natural way, producing two smaller spherical buildingsthat are canonically isomorphic to one another. In the processit is shown that to each wall and panel of these buildings atinfinity is attached a twin tree.     

Embeddings of Twin Trees

In this article we provide sufficient conditions for when a pair of trees having a semi-codistance function can be embedded in a twin tree with codistance function extending . We use these conditions to show that given a twin tree T of bidegree (d1,d2) there exists a twin tree of bidegree (e1,e2) containing T as a substructure as long as di ei.     

Property A and affine buildings

Yu's Property A is a non-equivariant generalisation of amenability introduced in his study of the coarse Baum Connes conjecture. In this paper we show that all affine buildings of type , and have Property A. Together with results of Guentner, Higson and Weinberger, this completes a programme to show that all affine building have Property A. In passing we use our technique to obtain a new proof for groups acting on buildings.     

Strongly transitive actions on euclidean buildings

We prove a decomposition result for a group G acting strongly transitively on the Tits boundary of a Euclidean building. As an application we provide a local to global result for discrete Euclidean buildings, which generalizes results in the locally compact case by Caprace–Ciobotaru and Burger–Mozes. Let X be a Euclidean building without cone factors. If a group G of automorphisms of X acts strongly transitively on the spherical building at infinity ?X, then the G-stabilizer of every affine apartment in X contains all reflections along thick walls. In particular G acts strongly transitively on X if X is simplicial and thick.     

Groups acting simply transitively on the vertices of a building of typeà 2, I

If a group acts simply transitively on the vertices of an affine building with connected diagram, then must be of typeà _n–1 for somen2, and must have a presentation of a simple type. The casen=2, when is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when is the building ofG=PGL(3,K),K a local field, and when G. We exhibit such a group whenK=F _q ((X)),q any prime power. Our study leads to combinatorial objects which we calltriangle presentations. These triangle presentations give rise to some new buildings of typeà ₂.     

Universal Covers of Geometries of Far Away Type

The geometries studied in this paper are obtained from buildings of spherical type by removing all chambers at non-maximal distance from a given element or flag. I consider a number of special cases of the above construction chosen among those which most frequently appear in the literature, proving that the resulting geometry is always simply connected but for three cases of small rank defined over GF(2) and GF(4). I also compute the universal cover in those exceptional cases.     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

An Elementary Approach to the Twin Primes Problem

Hardy-Littlewood [4] conjectured an asymptotic formula for the number of prime pairs (twin primes) (p, p+2d) with p+2dy, where d N is fixed and y . Up to now, no one has been able to prove this conjecture, but employing Hardy-Littlewoods circle method, Lavrik [5] showed that in a certain sense this formula holds true for almost-all dy/2.In the present paper, we use a completely different method to prove Lavriks almost-all result. Our method is based on an elementary approach developed by Pan Chengdong [7] to the twin primes problem. By a slight modification of our method, we get a corresponding almost-all result for the binary Goldbach problem. From this, according to [3], we derive Vinogradovs [8] well-known Three-Primes-Theorem.     

The geometry of the chamber system of a semimodular lattice

In this paper geometric properties of the following metric space C are studied. Its elements are called chambers and are the maximal chains of a semimodular lattice X of finite height and its metric d is the gallery distance. We show that X has many properties in common with buildings. More specifically, Tits [17] has recently described buildings in terms of Weyl-group valued distance functions. We consider the Jordan-Hölder permutation (C, D) corresponding to a pair C, D of chambers and show that it has most properties of such a distance with values in the symmetric group.     

Sur les immeubles hyperboliques (On hyperbolic buildings)

The aim of this paper is to give a geometric approach to Tits' amalgam method to construct buildings and to initiate a study of hyperbolic buildings, i.e. whose types are reflexion systems of the real hyperbolic space. We construct lots of examples and study their cohomology at infinity. We construct CAT(–1) polyhedral complexes having big discrete parabolic groups of isometries.     

Symplectic polarities of buildings of type E 6

A symplectic polarity of a building Δ of type E ₆ is a polarity whose fixed point structure is a building of type F ₄ containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality θ of Δ never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then θ is a symplectic duality. Secondly, we show that, if a duality θ never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E ₆ of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E ₆ for which the Phan geometry is empty.     

Twin positive solutions of boundary value problems for finite difference equations with p-Laplacian operator

In this paper, a new twin fixed point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem

     

Twin trees and -gons

We define a natural generalization of generalized -gons to the case of -graphs (where is a totally ordered abelian group and ). We term these objects -gons. We then show that twin trees as defined by Ronan and Tits can be viewed as -gons, where is ordered lexicographically. This allows us to then generalize twin trees to the case of -trees. Finally, we give a free construction of -gons in the cases where is discrete and has a subgroup of index that does not contain the minimal element of .

     

Spherical harmonic analysis on affine buildings

Let be a locally finite regular affine building with root system R. There is a commutative algebra spanned by averaging operators A _λ , λ ∈ P ⁺, acting on the space of all functions f:V _P → , where V _P is in most cases the set of all special vertices of , and P ⁺ is a set of dominant coweights of R. This algebra is studied in [6] and [7] for à _n buildings, and the general case is treated in [15]. In this paper we show that all algebra homomorphisms h: may be expressed in terms of the Macdonald spherical functions. We also provide a second formula for these homomorphisms in terms of an integral over the boundary of . We may regard as a subalgebra of the C ^*-algebra of bounded linear operators on ?²(V _P ), and we write for the closure of in this algebra. We study the Gelfand map , where M ₂= , and we compute M ₂ and the Plancherel measure of . We also compute the ?²-operator norms of the operators A _λ , λ ∈ P ⁺, in terms of the Macdonald spherical functions.     

Characterizations by Automorphism Groups of some Rank 3 Buildings – I. Some Properties of Half Strongly-Transitive Triangle Buildings

In a sequence of papers, we will show that the existence of a (half) strongly-transitive automorphism group acting on a locally finite triangle building forces to be one of the examples arising from PSL₃(K) for a locally finite local skewfield K. Furthermore, we introduce some Moufang-like conditions in affine buildings of rank 3, and characterize those examples arising from algebraic, classical or mixed type groups over a local field. In particular, we characterize the p-adic-like affine rank 3 buildings by a certain p-adic Moufang condition, and show that such a condition has zero probability to survive in hyperbolic rank 3 buildings. This shows that a construction of hyperbolic buildings as analogues of p-adic affine buildings is very unlikely to exist.     

Groups acting simply transitively on the vertices of a building of typeà 2, II: The casesq=2 andq=3

If (P, L) is a projective plane and is a triangle presentation compatible with a point-line correspondence :P L, then gives rise to a group and a thick building of typeà ₂ on the vertices of which acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence :P L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these , is isomorphic to the building associated withG=PGL(3,K) whereK is a local field with discrete valuation and residual field of orderq. We identify the for which this is the case, and in these cases, find embeddings of intoG. We also describe the arithmetic nature of these groups.     

On epimorphisms of spherical Moufang buildings

In this paper, we classify the epimorphisms of irreducible spherical Moufang buildings (of rank $\ge 2$) defined over a field. As an application, we characterize indecomposable epimorphisms of these buildings as those epimorphisms arising from $\mathbb{R}$-buildings.