Affine Building Of Type Research Articles

Sharp estimates for distinguished random walks on affine buildings of type [formula omitted]

We study a distinguished random walk on affine buildings of type A˜r , which was already considered by Cartwright, Saloff-Coste and Woess. In rank r=2, it is the simple random walk and we obtain optimal global bounds for its transition density (same upper and lower bound, up to multiplicative constants). In the higher rank case, we obtain sharp uniform bounds in fairly large space–time regions which are sufficient for most applications.

Boundary behavior of generalized Poisson integrals on buildings of type $$\widetilde{A}_2$$ A ~ 2

Let $${\Delta }$$ be an affine building of type $$\widetilde{A}_2$$ and $${\Omega }$$ its maximal boundary. We prove that, for every function $$f \in L^1({\Omega }),$$ restricted convergence of the normalized generalized Poisson transform $${\mathcal {P}}_{{\chi }} f / {\mathcal {P}}_{{\chi }} 1$$ to f holds almost everywhere.

AN EIGENSPACE OF LARGE DIMENSION FOR A HECKE ALGEBRA ON AN BUILDING

AbstractLet Δ be an affine building of type $\tilde A_2$ and let 𝔸 be its fundamental apartment. We consider the set 𝕌0 of vertices of type 0 of 𝔸 and prove that the Hecke algebra of all W0-invariant difference operators with constant coefficients acting on 𝕌0 has three generators. This property leads us to define three Laplace operators on vertices of type 0 of Δ. We prove that there exists a joint eigenspace of these operators having dimension greater than ∣W0 ∣. This implies that there exist joint eigenfunctions of the Laplacians that cannot be expressed, via the Poisson transform, in terms of a finitely additive measure on the maximal boundary Ω of Δ.

On the existence of certain affine buildings of type E 6 and E 7

We show that every Moufang quadrangle of type E6 or E7 defined over a field complete with respect to a discrete valuation is the building at infinity of an affine building.

Generalized polygons with non-discrete valuation defined by two-dimensional affine ℝ-buildings

Abstract In this paper, we show that the building at infinity of a two-dimensional affine ℝ-building is a generalized polygon endowed with a valuation satisfying some specific axioms. Specializing to the discrete case of affine buildings, this solves part of a long standing conjecture about affine buildings of type , and it reproves the results obtained mainly by the second author for types and . The techniques are completely different from the ones employed in the discrete case, but they are considerably shorter, and general (i.e., independent of the type of the two-dimensional ℝ-building).

Random walk on a building of type Ãr and Brownian motion of the Weyl chamber

In this paper we study a random walk on an affine building of type $\tilde{A}_r$, whose radial part, when suitably normalized, converges to the Brownian motion of the Weyl chamber. This gives a new discrete approximation of this process, alternative to the one of Biane \cite{Bia2}. This extends also the link at the probabilistic level between Riemannian symmetric spaces of the noncompact type and their discrete counterpart, which had been previously discovered by Bougerol and Jeulin in rank one \cite{BJ}. The main ingredients of the proof are a combinatorial formula on the building and the estimate of the transition density proved in \cite{AST}.

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 A ˜ 2 , B ˜ 2 and G ˜ 2 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 A ˜ n buildings.

The reduced group C*-algebra of a triangle building

Let Δ be an affine building of type Ã2 and let Γ be a discrete group of type-rotating automorphisms acting simply transitively on the vertices of δ. We prove that the reduced group C*-algebra is simple. To prove this result we use the sufficient condition for the simplicity of given in a recent paper by M. Bekka, M. Cowling and P. de la Harpe.

ENDOMORPHISMS OF DISCRETE GROUPS ACTING CHAMBER TRANSITIVELY ON AFFINE BUILDINGS

Let Γ be a group acting chamber transitively by type preserving automorphisms on a locally finite affine building of type Ã2. We show that Out(Γ) is finite and that Γ is Hopfian. We apply our results to affine Coxeter groups and a family of four groups discovered by J. Tits.

Generalized quadrangles with valuation

We show that the class of generalized quadranges with valuation (as defined in [13]) coincides with the class of the generalized quadrangles associated with the building at infinity of affine buildings of type \(\tilde C_2\) (up to duality).

The affine building of type� 2 over a local field of characteristic two

The affine building of type� 2 over a local field of characteristic two