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Alain Valette

Publications and source records attributed to Alain Valette.

At least 19 recordsLinked to original sources
arXiv2026

On the transportation cost norm on finite metric graphs

For a finite metric graph $X=(V,E,\ell)$, where $V$ is endowed with the shortest path metric, we consider the transportation cost problem associated with the distance $d$ on $V$. Namely, for $f$ a function with total sum 0 on $V$, write $f=\sum_{a,b\in V}P(a,b)(\delta_a-\delta_b)$ where the transportation plan $P$ satisfies $P(a,b)\geq 0$ for $(a,b)\in V\times V$. The cost of $P$ is $W(P):=\sum_{a,b\in V}P(a,b)d(a,b)$ and the transportation norm of $f$ is $\|f\|_{TC}=\min_P W(P)$ where $P$ runs over all transportation plans for $f$. In this semi-survey paper, we give short proofs for the following statements: 1)There always exists an optimal transportation plan supported in $V_+\times V_-$ where $V_+=\{x\in V: f(x)>0\}$ and $V_-=\{x\in V: f(x)<0\}$. If $X$ is a metric tree, we may moreover assume that this plan involves at most $|Supp(f)|-1$ transports. 2) There always exists an optimal transportation plan supported in the set of edges of $X$. 3) Better, there always exists an optimal transportation plan supported in some spanning tree of $X$. We use this to reprove known formulae for the transportation norm when $X$ is either a tree or a cycle.

math.MG↗
arXiv2025

Rieffel projections and 2-by-2 matrices

For a compact space $Y$, we view $C(Y\times S^1)$ as the crossed product $C(Y)\rtimes\mathbb{Z}$, with $\mathbb{Z}$ acting trivially. This allows us to study Rieffel projections in $M_2(C(Y\times S^1))$: we characterize them and compute their image under the projection $\partial_0:K_0(C(Y\times S^1))\rightarrow K_1(C(Y))$. We provide a new Rieffel projection in $M_2(C(\mathbb{T}^2))$, different from Loring's one, and involving only trigonometric polynomials plus the square root of $2-e^{2\pi i\theta}-e^{-2\pi i\theta}$. We give applications of this projection, e.g. explicit generators for the K-theory of $C(\mathbb{T}^3)$. Finally, we prove that, if a Banach algebra completion $\mathcal{B}$ of $\mathbb{C}[\mathbb{Z}^n]$ is continuously contained in $C(\mathbb{T}^n)$ and such that the Fourier series of $(2-e^{2\pi i\theta_j}-e^{-2\pi i\theta_j})^{1/2}\;(j=1,...,n)$ converges in $\mathcal{B}$, then the inclusion $\mathcal{B}\hookrightarrow C(\mathbb{T}^n)$ induces isomorphisms in K-theory.

math.KT↗
arXiv2023

The linear $\SL_2(\Z)$-action on $\T^n$: ergodic and von Neumann algebraic aspects

The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$ for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote by $G_n$ the group $\SL_2(\Z)$ for $n$ even, and $\PSL_2(\Z)$ for $n$ odd. We prove that the action is free and ergodic for every $n\geq 2$, that if $h\in \SL_2(\Z)$ is a hyperbolic element and if $n$ is even, then the action of the subgroup generated by $h$ is still ergodic, but also that, for $n$ odd, no amenable subgroup of $\PSL_2(\Z)$ acts ergodically on $\T^n$. We deduce also that every ergodic sub-equivalence relation $\Rr$ of the orbital equivalence relation $\mathcal{S}_n$ of $G_n$ on $\T^n$ is either amenable or rigid, extending a result by Ioana for $n=2$. This result has the following corollaries: firstly, for $n\geq 2$ even, if $H$ is a maximal amenable subgroup of $\SL_2(\Z)$ containing an hyperbolic matrix, then the associated crossed product II$_1$ factor $L^\infty(\T^n)\rtimes H$ is a maximal Haagerup subalgebra of $L^\infty(\T^n)\rtimes \SL_2(\Z)$; secondly , for every $n$, the fundamental group of $L^\infty(\T^n)\rtimes G_n$ is trivial.

math.OA↗
arXiv2023

Maximal Haagerup subgroups in $\mathbb{Z}^{n+1}\rtimes_{\rho_n}GL_2(\mathbb{Z})$

For $n\geq 1$, let $\rho_n$ denote the standard action of $GL_2(\Z)$ on the space $P_n(\Z)\simeq\Z^{n+1}$ of homogeneous polynomials of degree $n$ in two variables, with integer coefficients. For $G$ a non-amenable subgroup of $GL_2(\Z)$, we describe the maximal Haagerup subgroups of the semi-direct product $\Z^{n+1}\rtimes_{\rho_n} G$, extending the classification of Jiang-Skalski \cite{JiSk} of the maximal Haagerup subgroups in $\Z^2\rtimes SL_2(\Z)$. We prove that, for $n$ odd, the group $P_n(\Z)\rtimes SL_2(\Z)$ admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are free groups; and that, for $n$ even, the group $P_n(\Z)\rtimes GL_2(\Z)$ admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are isomorphic to $SL_2(\Z)$.

math.GR↗
arXiv2022

Equivariant K-homology and K-theory for some discrete planar affine groups

We consider the semi-direct products $G=\mathbb Z^2\rtimes GL_2(\mathbb Z), \mathbb Z^2\rtimes SL_2(\mathbb Z)$ and $\mathbb Z^2\rtimes\Gamma(2)$ (where $\Gamma(2)$ is the congruence subgroup of level 2). For each of them, we compute both sides of the Baum-Connes conjecture, namely the equivariant $K$-homology of the classifying space $\underline{E}G$ for proper actions on the left-hand side, and the analytical K-theory of the reduced group $C^*$-algebra on the right-hand side. The computation of the LHS is made possible by the existence of a 3-dimensional model for $\underline{E}G$, which allows to replace equivariant K-homology by Bredon homology. We pay due attention to the presence of torsion in $G$, leading to an extensive study of the wallpaper groups associated with finite subgroups. For the second and third groups, the computations in $K_0$ provide explicit generators that are matched by the Baum-Connes assembly map.

math.OA↗
arXiv2022

The exceptional simple Lie group $F_{4(-20)}$, after J. Tits

This is a semi-survey paper, where we start by advertising Tits' synthetic construction from \cite{Tits}, of the hyperbolic plane $H^2(Cay)$ over the Cayley numbers $Cay$, and of its automorphism group which is the exceptional simple Lie group $G=F_{4(-20)}$. Let $G=KAN$ be the Iwasawa decomposition. Our contributions are: a) Writing down explicitly the action of $N$ on $H^2(Cay)$ in Tits'model, facing the lack of associativity of $Cay$. b) If $MAN$ denotes the minimal parabolic subgroup of $G$, characterizing $M$ geometrically.

math.GR↗
arXiv2021

Wasserstein distance and metric trees

We study the Wasserstein (or earthmover) metric on the space $P(X)$ of probability measures on a metric space $X$. We show that, if a finite metric space $X$ embeds stochastically with distortion $D$ in a family of finite metric trees, then $P(X)$ embeds bi-Lipschitz into $\ell^1$ with distortion $D$. Next, we re-visit the closed formula for the Wasserstein metric on finite metric trees due to Evans-Matsen \cite{EvMat}. We advocate that the right framework for this formula is real trees, and we give two proofs of extensions of this formula: one making the link with Lipschitz-free spaces from Banach space theory, the other one algorithmic (after reduction to finite metric trees).

math.MG↗
arXiv2021

On arithmetic properties of solvable Baumslag-Solitar groups

For $0<\alpha\le 1$, we say that a sequence $(X_k)_{k>0}$ of $d$-regular graphs has property $D_\alpha$ if there exists a constant $C>0$ such that $\mathrm{diam}(X_k)\ge C\cdot|X_k|^\alpha$. We investigate property $D_\alpha$ for arithmetic box spaces of the solvable Baumslag-Solitar groups $BS(1,m)$ (with $m\geq 2$): those are box spaces obtained by embedding $BS(1,m)$ into the upper triangular matrices in $GL_2(\mathbb{Z}[1/m])$ and intersecting with a family $M_{N_k}$ of congruence subgroups of $GL_2(\mathbb{Z}[1/m])$, where the levels $N_k$ are coprime with $m$ and $N_k|N_{k+1}$. We prove: - if an arithmetic box space has $D_\alpha$, then $\alpha\le\frac{1}{2}$~; - if the family $(N_k)_k$ of levels is supported on finitely many primes, the corresponding arithmetic box space has $D_{1/2}$~; - if the family $(N_k)_k$ of levels is supported on a family of primes with positive analytic primitive density, then the corresponding arithmetic box space does not have $D_\alpha$, for every $\alpha>0$. Moreover, we prove that if we embed $BS(1,m)$ in the group of invertible upper-triangular matrices $T_n(\mathbb{Z}[1/m])$, then every finite index subgroup of the embedding contains a congruence subgroup. This is a version of the congruence subgroup property (CSP).

math.GR↗
arXiv2019

The Chabauty space of $\mathbb{Q}_p^\times$

Let $\mathcal{C}(G)$ denote the Chabauty space of closed subgroups of the locally compact group $G$. In this paper, we first prove that $\mathcal{C} (\mathbb{Q}_p^\times)$ is a proper compactification of $\mathbb{N}$, identified with the set $N$ of open subgroups with finite index. Then we identify the space $\mathcal{C}(\mathbb{Q}_p^\times) \smallsetminus N$ up to homeomorphism: e.g. for $p=2$, it is the Cantor space on which 2 copies of $\overline{\mathbb{N}}$ (the 1-point compactification of $\mathbb{N}$) are glued.

math.GN↗
arXiv2017

Property (T), finite-dimensional representations, and generic representations

Let $G$ be a discrete group with property (T). It is a standard fact that, in a unitary representation of $G$ on a Hilbert space $\mathcal{H}$, almost invariant vectors are close to invariant vectors, in a quantitative way. We begin by showing that, if a unitary representation has some vector whose coefficient function is close to a coefficient function of some finite-dimensional unitary representation $\sigma$, then the vector is close to a sub-representation isomorphic to $\sigma$: this makes quantitative a result of P.S. Wang [Wa]. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot [KLP], that a group $G$ with property (T) and such that $C^*(G)$ is residually finite-dimensional, admits a unitary representation which is generic (i.e. the orbit of this representation in $\mathrm{Rep}(G,\mathcal{H})$ under the unitary group $U(\mathcal{H})$ is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in $\mathrm{Rep}(G,\mathcal{H})$.

math.GR↗
arXiv2017

Chabauty Limits of Subgroups of $SL(n, \mathbb{Q}_p)$

We study the Chabauty compactification of two families of closed subgroups of $SL(n,\mathbb{Q}_p)$. The first family is the set of all parahoric subgroups of $SL(n,\mathbb{Q}_p)$. Although the Chabauty compactification of parahoric subgroups is well studied, we give a different and more geometric proof using various Levi decompositions of $SL(n,\mathbb{Q}_p)$. Let $C$ be the subgroup of diagonal matrices in $SL(n, \mathbb{Q}_p)$. The second family is the set of all $SL(n,\mathbb{Q}_p)$-conjugates of $C$. We give a classification of the Chabauty limits of conjugates of $C$ using the action of $SL(n,\mathbb{Q}_p)$ on its associated Bruhat--Tits building and compute all of the limits for $n\leq 4$ (up to conjugacy). In contrast, for $n\geq 7$ we prove there are infinitely many $SL(n,\mathbb{Q}_p)$-nonconjugate Chabauty limits of conjugates of $C$. Along the way we construct an explicit homeomorphism between the Chabauty compactification in $\mathfrak{sl}(n, \mathbb{Q}_p)$ of $SL(n,\mathbb{Q}_p)$-conjugates of the $p$-adic Lie algebra of $C$ and the Chabauty compactification of $SL(n,\mathbb{Q}_p)$-conjugates of $C$.

math.GT↗
arXiv2017

Locally compact groups with every isometric action bounded or proper

A locally compact group $G$ has property PL if every isometric $G$-action either has bounded orbits or is (metrically) proper. For $p>1$, say that $G$ has property $BP_{L^p}$ if the same alternative holds for the smaller class of affine isometric actions on $L^p$-spaces. We explore properties PL and $BP_{L^p}$ and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix by Corina Ciobotaru provides new examples of groups with property PL, including non-linear ones.

math.GR↗
arXiv2016

K-homology and K-theory for the lamplighter groups of finite groups

Let $F$ be a finite group. We consider the lamplighter group $L=F\wr\mathbb{Z}$ over $F$. We prove that $L$ has a classifying space for proper actions $\underline{E} L$ which is a complex of dimension two. We use this to give an explicit proof of the Baum-Connes conjecture (without coefficients), that states that the assembly map $\mu_i^L:K_i^L(\underline{E} L)\rightarrow K_i(C^*L)\;(i=0,1)$ is an isomorphism. Actually, $K_0(C^*L)$ is free abelian of countable rank, with an explicit basis consisting of projections in $C^*L$, while $K_1(C^*L)$ is infinite cyclic, generated by the unitary of $C^*L$ implementing the shift. Finally we show that, for $F$ abelian, the $C^*$-algebra $C^*L$ is completely characterized by $|F|$ up to isomorphism.

math.OA↗
arXiv2016

K-theory for the $C^*$-algebras of the solvable Baumslag-Solitar groups

We provide a new computation of the K-theory of the group $C^*$-algebra of the solvable Baumslag-Solitar group $BS(1,n)\;(n\neq 1)$; our computation is based on the Pimsner-Voiculescu 6-terms exact sequence, by viewing $BS(1,n)$ as a semi-direct product $\mathbb{Z}[1/n]\rtimes\mathbb{Z}$. We deduce from it a new proof of the Baum-Connes conjecture with trivial coefficients for $BS(1,n)$.

math.OA↗
arXiv2015

Expanders and box spaces

We consider box spaces of finitely generated, residually finite groups $G$, and try to distinguish them up to coarse equivalence. We show that, for $n\geq 2$, the group $SL_n(\mathbb{Z})$ has a continuum of box spaces which are pairwise non-coarsely equivalent expanders. Moreover, varying the integer $n\geq 3$, expanders given as box spaces of $SL_n(\mathbb{Z})$ are pairwise inequivalent; similarly, varying the prime $p$, expanders given as box spaces of $SL_2(\mathbb{Z}[\sqrt{p}])$ are pairwise inequivalent. A strong form of non-expansion for a box space is the existence of $α\in]0,1]$ such that the diameter of each component $X_n$ satisfies $diam(X_n)=Ω(|X_n|^α)$. By a result of Breuillard and Tointon, the existence of such a box space implies that $G$ virtually maps onto $\mathbb{Z}$: we establish the converse. For the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ and for a semi-direct product $\mathbb{Z}^2\rtimes\mathbb{Z}$, such box spaces are explicitly constructed using specific congruence subgroups. We finally introduce the full box space of $G$, i.e. the coarse disjoint union of all finite quotients of $G$. We prove that the full box space of a group mapping onto the free group $\mathbb{F}_2$ is not coarsely equivalent to the full box space of an $S$-arithmetic group satisfying the Congruence Subgroup Property.

math.GR↗
arXiv2014

The Mayer-Vietoris Sequence for Graphs of Groups, Property (T), and the First $\ell^2$-Betti Number

We explore the Mayer-Vietoris sequence developed by Chiswell for the fundamental group of a graph of groups when vertex groups satisfy some vanishing assumption on the first cohomology (e.g. property (T), or vanishing of the first $\ell^2$-Betti number). We characterize the vanishing of first reduced cohomology of unitary representations when vertex stabilizer have property (T). We find necessary and sufficient conditions for the vanishing of the first $\ell^2$-Betti number. We also study the associated Haagerup cocycle and show that it vanishes in first reduced cohomology precisely when the action is elementary.

math.GR↗
arXiv2014

Irreducible affine isometric actions on Hilbert spaces

We undertake a systematic study of irreducible affine isometric actions of locally compact groups on Hilbert spaces. It turns out that, while that are a few parallels of this study to the by now classical theory of irreducible unitary representations, these two theories differ in several aspects (for instance, the direct sum of two irreducible affine actions can still be irreducible). One of the main tools we use is an affine version of Schur's lemma characterizing the irreducibility of an affine isometric group action. This enables us to describe for instance the irreducible affine isometric actions of nilpotent groups. As another application, a short proof is provided for the following result of Neretin: the restriction to a cocompact lattice of an irreducible affine action of locally compact group remains irreducible. We give a necessary and sufficient condition for a fixed unitary representation to be the linear part of an irreducible affine action. In particular, when the unitary representation is a multiple of the regular representation of a discrete group G, we show how this question is related to the L2-Betti number of G. After giving a necessary and sufficient condition for a direct sum of irreducible affine actions to be irreducible, we show the following super-rigidity result: if G is product of two or more locally compact groups, then every irreducible affine action of any irreducible co-compact lattice in G extends to an affine action of G, provided the linear part of this action does not weakly contain the trivial representation.

math.GR↗