arXiv subjects
Publications and source records attributed to Bertrand Rémy.
This article deals with harmonic analysis on affine buildings. Its main goal is to construct suitable kernels associated to a discrete multitemporal wave equations on the latter spaces, the long-standing motivation being to contribute to progress in arithmetic quantum unique ergodicity (AQUE) on certain Riemannian manifolds.
We carry out an in-depth study of Martin compactifications of affine buildings, from the viewpoint of potential theory and random walks. This work does not use any group action on buildings, although all the results are also stated within the framework of the Bruhat--Tits theory of semisimple groups over non-Archimedean local fields. This choice should allow the use of these building compactifications in intriguing geometric group theory situations, where only lattice actions are available. The resulting compactified spaces use and, at the same time, make it possible to understand geometrically the descriptions of asymptotic behavior of kernels resulting from the non-Archimedean harmonic analysis on affine buildings. Along the paper, we make explicit the most substantial differences with the case of symmetric spaces, namely absence of a group action but existence of precise asymptotics of Green kernels and, of course, no possibility to stand by standard techniques from PDEs.
We initiate the investigation of critical exponents (in degree equal to the rank) for the vanishing of L^p-cohomology of higher rank Lie groups and related manifolds. We deal with the rank 2 case and exhibit such phenomena for SL$_3$(R) and for a family of 5-dimensional solvable Lie groups. This leads us to exhibit a continuum of quasi-isometry classes of rank 2 solvable Lie groups of non-positive curvature. We provide a detailed description of the $L^p$-cohomology of the real and complex hyperbolic spaces, to be combined with a spectral sequence argument for our higher-rank results.
We obtain non-vanishing of group $L^p$-cohomology of Lie groups for $p$ large and when the degree is equal to the rank of the group. This applies both to semisimple and to some suitable solvable groups. In particular, it confirms that Gromov's question on vanishing below the rank is formulated optimally. To achieve this, some complementary vanishings are combined with the use of spectral sequences. To deduce the semisimple case from the solvable one, we also need comparison results between various theories for $L^p$-cohomology, allowing the use of quasi-isometry invariance.
Given a semisimple group over a complete non-Archimedean field, it is well known that techniques from non-Archimedean analytic geometry provide an embedding of the corresponding Bruhat-Tits builidng into the analytic space associated to the group; by composing the embedding with maps to suitable analytic proper spaces, this eventually leads to various compactifications of the building. In the present paper, we give an intrinsic characterization of this embedding.
We go back and forth between, on the one hand, presentations of arithmetic and Kac-Moody groups and, on the other hand, presentations of profinite groups, deducing along the way new results on both.
We prove simplicity for incomplete rank 2 Kac-Moody groups over algebraic closures of finite fields with trivial commutation relations between root groups corresponding to prenilpotent pairs. We don't use the (yet unknown) simplicity of the corresponding finitely generated groups (i.e., when the ground field is finite). Nevertheless we use the fact that the latter groups are just infinite (modulo center).
We investigate Bruhat-Tits buildings and their compactifications by means of Berkovich analytic geometry over complete non-Archimedean fields. For every reductive group G over a suitable non-Archimedean field k we define a map from the Bruhat-Tits building B(G,k) to the Berkovich analytic space Gan asscociated with G. Composing this map with the projection of G^an to its flag varieties, we define a family of compactifications of B(G,k). This generalizes results by Berkovich in the case of split groups. Moreover, we show that the boundary strata of the compactified buildings are precisely the Bruhat-Tits buildings associated with a certain class of parabolics. We also investigate the stabilizers of boundary points and prove a mixed Bruhat decomposition theorem for them.
Let GF denote the rational points of a semisimple group G over a non-archimedean local field F, with Bruhat-Tits building X. This paper contains five main results. We prove a convergence theorem for sequences of parahoric subgroups of GF in the Chabauty topology, which enables to compactify the vertices of X. We obtain a structure theorem showing that the Bruhat-Tits buildings of the Levi factors all lie in the boundary of the compactification. Then we obtain an identification theorem with the polyhedral compactification (previously defined in analogy with the case of symmetric spaces). We finally prove two parametrization theorems extending the BruhatTits dictionary between maximal compact subgroups and vertices of X: one is about Zariski connected amenable subgroups, and the other is about subgroups with distal adjoint action.