Anna Zappa

Groups acting simply transitively on the vertices of a building of typeÃ2, I

If a group Γ acts simply transitively on the vertices of an affine building Δ with connected diagram, then Δ must be of typeÃn−1 for somen⩾2, and Γ must have a presentation of a simple type. The casen=2, when Δ is a tree, has been studied in detail. We consider the casen=3, motivated particularly by the case when Δ is the building ofG=PGL(3,K),K a local field, and when Γ⩽G. We exhibit such a group Γ whenK=Fq((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 typeÃ2.

The Poisson transform and representations of a free group

Abstract Let G be a free group with r generators, 1 r G which plays the same role of the Laplace Beltrami operator on semisimple Lie groups are characterized. Furthermore, an analytic family of representations π z of G on functions on the boundary Ω is considered, defined by π z (x)ƒ(ω) = p z (x, ω)ƒ(x −1 ω) , where p ( x , ω ) is the Poisson kernel relative to the action of G on Ω. It is proved that, for 0 s = Re z π z is uniformly bounded on an appropriate Hilbert space H s (Ω) . Finally the uniform boundedness of other special representations of G , obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of GL (2, Q p ) is proved.

Groups acting simply transitively on the vertices of a building of typeÃ2, II: The casesq=2 andq=3

If (P, L) is a projective plane and ℐ is a ‘triangle presentation compatible with a point-line correspondence λ:P →L’, then ℐ gives rise to a group Γℐ and a thick building Δℐ of typeÃ2 on the vertices of which Γℐ acts simply transitively. We find all triangle presentations (up to natural equivalence) compatible with some point-line correspondence λ:P →L, when (P, L) is the projective plane of orderq=2 orq=3. For some, but not all, of these ℐ, Δℐ 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 ℐ for which this is the case, and in these cases, find embeddings of Γℐ intoG. We also describe the arithmetic nature of these groups.

Eigenfunctions of the Laplace operators for a building of type\(\tilde A_2 \)

Let Δ be a thick affine building of type\(\tilde A_2 \) and of order q. We prove that each eigenfunction of the Laplace operators of Δ is the Poisson transform of a suitable finitely additive measure on the maximal boundary Ω.AbstractLet Δ be a thick affine building of type % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyqayaaia% WaaSbaaSqaaiaaikdaaeqaaaaa!37AA!

Eigenvalues of the Vertex Set Hecke Algebra of an Affine Building

\tilde A_2

Uniformly bounded representations and LP- convolution operators on a free group

and of order q. We prove that each eigenfunction of the Laplace operators of Δ is the Poisson transform of a suitable finitely additive measure on the maximal boundary Ω.

Irreducibility of the analytic continuation of the principal series of a free group

SummaryThe spherical principal series of a non-commutative free group may be analytically continued to yield a series of uniformly bounded representations, much as the spherical representations π(in1/2) + it of SL (2,R) may be analytically continued in the strip 0 < Rez < 1. This series of uniformly bounded representations was constructed and studied by A. M.Mantero and A.Zappa. Independently T.Pytlik and R.Szwarc introduced and studied representations of the free group which contain a series of subrepresentations indexed by spherical functions. Both series consist of irreducible representations and include the spherical complementary series. The aim of this paper is to prove that the non-unitary uniformly bounded representations of the two series are also equivalent.

Boundary behaviour of Poisson integrals on buildings of type Ã2

We study the eigenvalues of the vertex set Hecke algebra of an affine building, and prove, by a direct approach, the Weyl group invariance of any eigenvalue associated to a character. Moreover, we construct the Satake isomorphism of the Hecke algebra and we prove, by this isomorphism, that every eigenvalue arises from a character.