Alain Valette | University of Neuchâtel (original) (raw)

Papers by Alain Valette

Cambridge University Press eBooks, Apr 17, 2008

arXiv (Cornell University), May 24, 2019

To Alain Connes, for providing lifelong inspiration * (G), is called the geometric, or topologica... more To Alain Connes, for providing lifelong inspiration * (G), is called the geometric, or topological side. This is actually misleading, as its definition is awfully analytic, involving Kasparov's bivariant theory (see Chapter 3). A better terminology would be the commutative side, as indeed it involves a space EG, the classifiying space for proper actions of G (see Chapter 4), and K top * (G) is the G-equivariant K-homology of EG. When G is discrete and torsion-free, then EG = EG = BG, the universal cover of the classifying space BG. As G acts freely on EG, the G-equivariant K-homology of EG is K * (BG), the ordinary K-homology of BG, where Khomology for spaces can be defined as the homology theory dual to topological K-theory for spaces. • The assembly map µ r will be defined in Chapter 4 using Kasparov's equivariant KK-theory. Let us only give here a flavor of the meaning of this map. It was discovered in the late 1970's and early 1980's that the K-theory group K * (C * r (G)) is a receptacle for indices, see section 2.3. More precisely, if M is a smooth manifold with a proper action of G and compact quotient, and D an elliptic G-invariant differential operator on M , then D has an index ind G (D) living in K * (C * r (G)). Therefore, the geometric group K top * (G) should be thought of as the set of homotopy classes of such pairs (M, D), and the assembly map µ r maps the class [(M, D)] to ind G (D) ∈ K * (C * r (G)).

Cambridge University Press eBooks, Apr 17, 2008

Notices of the American Mathematical Society, 2023

Jacques Tits was born in Uccle, a municipality of Brussels, on August 12, 1930, and died on Decem... more Jacques Tits was born in Uccle, a municipality of Brussels, on August 12, 1930, and died on December 5, 2021. The son of a mathematician, Tits displayed extraordinary mathematical ability at an early age. He received his doctorate at the University of Brussels in 1950 and spent the following year at the Institute for Advanced Study. In 1964, he moved from the University of Brussels to a professorship in Bonn, and then in 1973 to the Collège de France, where he remained for the rest of his career. For almost thirty years he held courses and seminars at the Collège de France and for nineteen years, Tits was editor-in-chief of the Publications Mathématiques de l'IHES. Tits made many fundamental contributions to our understanding of the structure of semisimple algebraic groups and finite simple groups and did more than anyone to explore and reveal the geometric nature of these subjects. When Tits was young, Chevalley had shown that semisimple algebraic groups over an algebraically closed

Journal of Algebra, Apr 1, 2022

arXiv (Cornell University), Mar 11, 2023

For n ≥ 1, let ρ n denote the standard action of GL 2 (Z) on the space P n (Z) ≃ Z n+1 of homogen... more For n ≥ 1, let ρ n denote the standard action of GL 2 (Z) on the space P n (Z) ≃ 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 ⋊ ρn G, extending the classification of Jiang-Skalski [JS21] of the maximal Haagerup subgroups in Z 2 ⋊ SL 2 (Z). We prove that, for n odd, the group P n (Z) ⋊ 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) ⋊ GL 2 (Z) admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are isomorphic to SL 2 (Z).

arXiv (Cornell University), Mar 18, 2013

For ϕ a normalized positive definite function on a locally compact abelian group G, we consider o... more For ϕ a normalized positive definite function on a locally compact abelian group G, we consider on the one hand the unitary representation π ϕ associated to ϕ by the GNS construction, on the other hand the probability measure µ ϕ on the Pontryagin dualĜ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology H 1 (G, π ϕ) and reduced 1-cohomology H 1 (G, π ϕ). For example, H 1 (G, π ϕ) = 0 if and only if either Hom(G, C) = 0 or µ ϕ (1 G) = 0, where 1 G is the trivial character of G. * Partially supported by a ThinkSwiss Research Scholarship. 1 Recall Shalom's results, see Theorems 0.2 and 6.1 in [Sha00]: for a compactly generated group G, the group G has property (T), if and only if H 1 (G, π) = 0 for every unitary representation π of G, if and only if H 1 (G, σ) = 0 for every unitary irreducible representation of G.

arXiv (Cornell University), May 25, 2009

We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert s... more We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also provides a characterization of subsets with relative Property T in a standard wreath product.

arXiv (Cornell University), Jul 1, 2013

We compute Ä ¾-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordin... more We compute Ä ¾-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the Plancherel measure. This allows us to compute the Ä ¾-Betti numbers for semi-simple Lie groups with finite center, simple algebraic groups over local fields, and automorphism groups of locally finite trees acting transitively on the boundary.

Séminaire de théorie spectrale et géométrie, 1996

L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accor... more L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Cambridge University Press eBooks, 2001

arXiv (Cornell University), Nov 24, 2022

This is a semi-survey paper, where we start by advertising Tits' synthetic construction from [Ti5... more This is a semi-survey paper, where we start by advertising Tits' synthetic construction from [Ti53], 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: • Writing down explicitly the action of N on H 2 (Cay) in Tits'model, facing the lack of associativity of Cay. • If M AN denotes the minimal parabolic subgroup of G, characterizing M geometrically.

arXiv (Cornell University), Nov 13, 2017

Let G be a discrete group with property (T). It is a standard fact that, in a unitary representat... more 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 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 finitedimensional unitary representation σ, then the vector is close to a subrepresentation isomorphic to σ: this makes quantitative a result of P.S. Wang [12]. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot [9], 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 Rep(G, H) under the unitary group U (H) is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in Rep(G, H).

Involve, Mar 4, 2021

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

Cambridge University Press eBooks, 2001

arXiv (Cornell University), Dec 19, 2022

We consider the semi-direct products G = Z 2 GL 2 (Z), Z 2 SL 2 (Z) and Z 2 Γ(2) (where Γ(2) is t... more We consider the semi-direct products G = Z 2 GL 2 (Z), Z 2 SL 2 (Z) and Z 2 Γ(2) (where Γ(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 EG 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 EG, 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.

Cambridge University Press eBooks, Apr 17, 2008

arXiv (Cornell University), May 24, 2019

To Alain Connes, for providing lifelong inspiration * (G), is called the geometric, or topologica... more To Alain Connes, for providing lifelong inspiration * (G), is called the geometric, or topological side. This is actually misleading, as its definition is awfully analytic, involving Kasparov's bivariant theory (see Chapter 3). A better terminology would be the commutative side, as indeed it involves a space EG, the classifiying space for proper actions of G (see Chapter 4), and K top * (G) is the G-equivariant K-homology of EG. When G is discrete and torsion-free, then EG = EG = BG, the universal cover of the classifying space BG. As G acts freely on EG, the G-equivariant K-homology of EG is K * (BG), the ordinary K-homology of BG, where Khomology for spaces can be defined as the homology theory dual to topological K-theory for spaces. • The assembly map µ r will be defined in Chapter 4 using Kasparov's equivariant KK-theory. Let us only give here a flavor of the meaning of this map. It was discovered in the late 1970's and early 1980's that the K-theory group K * (C * r (G)) is a receptacle for indices, see section 2.3. More precisely, if M is a smooth manifold with a proper action of G and compact quotient, and D an elliptic G-invariant differential operator on M , then D has an index ind G (D) living in K * (C * r (G)). Therefore, the geometric group K top * (G) should be thought of as the set of homotopy classes of such pairs (M, D), and the assembly map µ r maps the class [(M, D)] to ind G (D) ∈ K * (C * r (G)).

Cambridge University Press eBooks, Apr 17, 2008

Notices of the American Mathematical Society, 2023

Jacques Tits was born in Uccle, a municipality of Brussels, on August 12, 1930, and died on Decem... more Jacques Tits was born in Uccle, a municipality of Brussels, on August 12, 1930, and died on December 5, 2021. The son of a mathematician, Tits displayed extraordinary mathematical ability at an early age. He received his doctorate at the University of Brussels in 1950 and spent the following year at the Institute for Advanced Study. In 1964, he moved from the University of Brussels to a professorship in Bonn, and then in 1973 to the Collège de France, where he remained for the rest of his career. For almost thirty years he held courses and seminars at the Collège de France and for nineteen years, Tits was editor-in-chief of the Publications Mathématiques de l'IHES. Tits made many fundamental contributions to our understanding of the structure of semisimple algebraic groups and finite simple groups and did more than anyone to explore and reveal the geometric nature of these subjects. When Tits was young, Chevalley had shown that semisimple algebraic groups over an algebraically closed

Journal of Algebra, Apr 1, 2022

arXiv (Cornell University), Mar 11, 2023

For n ≥ 1, let ρ n denote the standard action of GL 2 (Z) on the space P n (Z) ≃ Z n+1 of homogen... more For n ≥ 1, let ρ n denote the standard action of GL 2 (Z) on the space P n (Z) ≃ 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 ⋊ ρn G, extending the classification of Jiang-Skalski [JS21] of the maximal Haagerup subgroups in Z 2 ⋊ SL 2 (Z). We prove that, for n odd, the group P n (Z) ⋊ 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) ⋊ GL 2 (Z) admits infinitely many pairwise non-conjugate maximal Haagerup subgroups which are isomorphic to SL 2 (Z).

arXiv (Cornell University), Mar 18, 2013

For ϕ a normalized positive definite function on a locally compact abelian group G, we consider o... more For ϕ a normalized positive definite function on a locally compact abelian group G, we consider on the one hand the unitary representation π ϕ associated to ϕ by the GNS construction, on the other hand the probability measure µ ϕ on the Pontryagin dualĜ provided by Bochner's theorem. We give necessary and sufficient conditions for the vanishing of 1-cohomology H 1 (G, π ϕ) and reduced 1-cohomology H 1 (G, π ϕ). For example, H 1 (G, π ϕ) = 0 if and only if either Hom(G, C) = 0 or µ ϕ (1 G) = 0, where 1 G is the trivial character of G. * Partially supported by a ThinkSwiss Research Scholarship. 1 Recall Shalom's results, see Theorems 0.2 and 6.1 in [Sha00]: for a compactly generated group G, the group G has property (T), if and only if H 1 (G, π) = 0 for every unitary representation π of G, if and only if H 1 (G, σ) = 0 for every unitary irreducible representation of G.

arXiv (Cornell University), May 25, 2009

We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert s... more We study stability properties of the Haagerup property and of coarse embeddability in a Hilbert space, under certain semidirect products. In particular, we prove that they are stable under taking standard wreath products. Our construction also provides a characterization of subsets with relative Property T in a standard wreath product.

arXiv (Cornell University), Jul 1, 2013

We compute Ä ¾-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordin... more We compute Ä ¾-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the Plancherel measure. This allows us to compute the Ä ¾-Betti numbers for semi-simple Lie groups with finite center, simple algebraic groups over local fields, and automorphism groups of locally finite trees acting transitively on the boundary.

Séminaire de théorie spectrale et géométrie, 1996

L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accor... more L'accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Cambridge University Press eBooks, 2001

arXiv (Cornell University), Nov 24, 2022

This is a semi-survey paper, where we start by advertising Tits' synthetic construction from [Ti5... more This is a semi-survey paper, where we start by advertising Tits' synthetic construction from [Ti53], 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: • Writing down explicitly the action of N on H 2 (Cay) in Tits'model, facing the lack of associativity of Cay. • If M AN denotes the minimal parabolic subgroup of G, characterizing M geometrically.

arXiv (Cornell University), Nov 13, 2017

Let G be a discrete group with property (T). It is a standard fact that, in a unitary representat... more 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 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 finitedimensional unitary representation σ, then the vector is close to a subrepresentation isomorphic to σ: this makes quantitative a result of P.S. Wang [12]. We use that to give a new proof of a result by D. Kerr, H. Li and M. Pichot [9], 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 Rep(G, H) under the unitary group U (H) is comeager). We also show that, under the same assumptions, the set of representations equivalent to a Koopman representation is comeager in Rep(G, H).

Involve, Mar 4, 2021

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

Cambridge University Press eBooks, 2001

arXiv (Cornell University), Dec 19, 2022

We consider the semi-direct products G = Z 2 GL 2 (Z), Z 2 SL 2 (Z) and Z 2 Γ(2) (where Γ(2) is t... more We consider the semi-direct products G = Z 2 GL 2 (Z), Z 2 SL 2 (Z) and Z 2 Γ(2) (where Γ(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 EG 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 EG, 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.