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Bertrand Remy

Publications and source records attributed to Bertrand Remy.

15 recordsLinked to original sources
arXiv2021

Quasi-isometric invariance of continuous group $L^p$-cohomology, and first applications to vanishings

We show that the continuous $L^p$-cohomology of locally compact second countable groups is a quasi-isometric invariant. As an application, we prove partial results supporting a positive answer to a question asked by M.~Gromov, suggesting a classical behaviour of continuous $L^p$-cohomology of simple real Lie groups. In addition to quasi-isometric invariance, the ingredients are a spectral sequence argument and Pansu's vanishing results for real hyperbolic spaces. In the best adapted cases of simple Lie groups, we obtain nearly half of the relevant vanishings.

math.GR↗
arXiv2017

Wonderful compactifications of Bruhat-Tits buildings

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

math.GR↗
arXiv2015

Bruhat-Tits buildings and analytic geometry

This paper provides an overview of the theory of Bruhat-Tits buildings. Besides, we explain how Bruhat-Tits buildings can be realized inside Berkovich spaces. In this way, Berkovich analytic geometry canbe used to compactify buildings. We discuss in detail the example of the special linear group. Moreover, we give an intrinsic description of Bruhat-Tits buildings in the framework of non-Archimedean analytic geometry.

math.GR↗
arXiv2013

On some pro-p groups from infinite-dimensional Lie theory

We initiate the study of some pro-p-groups arising from infinite-dimensional Lie theory. These groups are completions of some subgroups of incomplete Kac-Moody groups over finite fields, with respect to various completions of algebraic or geometric origin. We show topological finite generation for the pro-p Sylow subgroups in many complete Kac-Moody groups. This implies abstract simplicity of the latter groups. We also discuss with the question of (non-)linearity of these pro-p groups.

math.GR↗
arXiv2009

On the distortion of twin building lattices

We show that twin building lattices are undistorted in their ambient group; equivalently, the orbit map of the lattice to the product of the associated twin buildings is a quasi-isometric embedding. As a consequence, we provide an estimate of the quasi-flat rank of these lattices, which implies that there are infinitely many quasi-isometry classes of finitely presented simple groups. In an appendix, we describe how non-distortion of lattices is related to the integrability of the structural cocycle.

math.GR↗
arXiv2009

Bruhat-Tits theory from Berkovich's point of view. II. Satake compactifications

In our previous paper "Bruhat-Tits theory from Berkovich's point of view. I ? Realizations and compactifications of buildings", we investigated realizations of the Bruhat-Tits building B(G,k) of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry, and we studied in detail the compactifications of the building which arise from this point of view. In this paper, we give a representation theoretic flavor to these compactifications, following Satake's original constructions for Riemannian symmetric spaces. We first prove that Berkovich compactifications of a building coincide with the compactifications previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding B(G,k) in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

math.GR↗
arXiv2007

Contraction groups in complete Kac-Moody groups

Let $G$ be an abstract Kac-Moody group over a finite field and $\bar{G}$ be the closure of the image of $G$ in the automorphism group of its positive building. We show that if the Dynkin diagram associated to $G$ is irreducible and neither of spherical nor of affine type, then the contraction groups of elements in $\bar{G}$ which are not topologically periodic are not closed. (In those groups there always exist elements which are not topologically periodic.)

math.GR↗
arXiv2006

Simplicity and superrigidity of twin building lattices

Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if and only if the corresponding buildings are (irreducible and) not of affine type (i.e. they are not Bruhat-Tits buildings). In fact, many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac-Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac-Moody group is always proper. We also show that Kac-Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices.

math.GR↗
arXiv2006

Flat rank of automorphism groups of buildings

The flat rank of a totally disconnected locally compact group G, denoted flat-rk(G), is an invariant of the topological group structure of G. It is defined thanks to a natural distance on the space of compact open subgroups of G. For a topological Kac-Moody group G with Weyl group W, we derive the inequalities: alg-rk(W)\le flat-rk(G)\le rk(|W|\_0). Here, alg-rk(W) is the maximal $\mathbb{Z}$-rank of abelian subgroups of W, and rk(|W|\_0) is the maximal dimension of isometrically embedded flats in the CAT0-realization |W|\_0. We can prove these inequalities under weaker assumptions. We also show that for any integer n \geq 1 there is a topologically simple, compactly generated, locally compact, totally disconnected group G, with flat-rk(G)=n and which is not linear.

math.GR↗
arXiv2004

Commensurators of some non-uniform tree lattices and Moufang twin trees

Sh. Mozes showed that the commensurator of the lattice ${\rm PSL}_2 \bigl({\bf F}_p[t{}^{-1}] \bigr)$ is dense in the full automorphism group of the Bruhat-Tits tree of valency $p+1$, the latter group being much bigger than ${\rm PSL}_2 \bigl({\bf F}_p((t)) \bigr)$. By G.A. Margulis' criterion, this density is a generalized arithmeticity result. We show that the density of the commensurator holds for many tree-lattices among those called of Nagao type by H. Bass and A. Lubotzky. The result covers many lattices obtained via Moufang twin trees.

math.GR↗
arXiv2004

Kac-Moody groups as discrete groups

This survey paper presents the discrete group viewpoint on Kac-Moody groups. Over finite fields, the latter groups are finitely generated; they act on new buildings enjoying remarkable negative curvature properties. The study of these groups is shared between proving results supporting the analogy with some $S$-arithmetic groups, and exhibiting properties showing that they are new groups.

math.GR↗
arXiv2003

Topological simplicity, commensurator super-rigidity and non-linearities of Kac-Moody groups

We provide new arguments to see topological Kac-Moody groups as generalized semisimple groups over local fields: they are products of topologically simple groups and their Iwahori subgroups are the normalizers of the pro-p Sylow subgroups. We use a dynamical characterization of parabolic subgroups to prove that some countable Kac-Moody groups with Fuchsian buildings are not linear. We show for this that the linearity of a countable Kac-Moody group implies the existence of a closed embedding of the corresponding topological group in a non-Archimedean simple Lie group, thanks to a commensurator super-rigidity theorem proved in the Appendix by P. Bonvin.

math.GR↗
arXiv2002

Kac-Moody groups: split and relative theories. Lattices

In this survey article, we recall some facts about split Kac-Moody groups as defined by J. Tits, describe their main properties and then propose an analogue of Borel-Tits theory for a non-split version of them. The main result is a Galois descent theorem, i.e. the persistence of a nice combinatorial structure after passing to rational points. We are also interested in the geometric point of view, namely the production of new buildings admitting (nonuniform) lattices.

math.GR↗
arXiv2002

Topological groups of Kac-Moody type, Fuchsian twinnings and their lattices

This paper deals with a class of totally disconnected groups acting on buildings, among which are certain Kac-Moody groups. The apartments of our buildings are hyperbolic planes tiled by right-angled polygons. We discuss linearity properties for the groups, as well as an analogy with semisimple groups over local fields of positive characteristic. Looking for counter-examples to this analogy leads to the construction of Moufang twinnings "with several ground fields".

math.GR↗