The trace class conjecture without the K-finiteness assumption

Let G be the group of real points of a reductive algebraic ℚ-group satisfying the same assumptions as in [5], Chapter I, and let Γ be a discrete subgroup of G. Let R_Γ be the right regular representation of G in L²(Γ\G). We prove in this Note that, for any integrable rapidly decreasing function ƒ on G, the restriction of R_Γ(ƒ) to the discrete spectrum of R_Γ is a trace class operator.     

A generalization of amenability and inner amenability of groups

Let G be a locally compact group. We continue our work [A. Ghaffari: Γ-amenability of locally compact groups, Acta Math. Sinica, English Series, 26 (2010), 2313–2324] in the study of Γ-amenability of a locally compact group G defined with respect to a closed subgroup Γ of G × G. In this paper, among other things, we introduce and study a closed subspace A _Γ ^p (G) of L ^∞(Γ) and then characterize the Γ-amenability of G using A _Γ ^p (G). Various necessary and sufficient conditions are found for a locally compact group to possess a Γ-invariant mean.     

On intransitive graph-restrictive permutation groups

Let Γ be a finite connected G-vertex-transitive graph and let v be a vertex of Γ. If the permutation group induced by the action of the vertex-stabiliser G _v on the neighbourhood Γ(v) is permutation isomorphic to L, then (Γ,G) is said to be locally L. A permutation group L is graph-restrictive if there exists a constant c(L) such that, for every locally L pair (Γ,G) and a vertex v of Γ, the inequality |G _v |≤c(L) holds. We show that an intransitive group is graph-restrictive if and only if it is semiregular.     

Construction d’entrelacs hyperpoliques et arithmétiques

In this Note, we continue the study started in [4] about arithmetic hyperbolic links L such that S³\L is homeomorphic to H³/Γ, where Γ is not conjugate to a subgroup of any Bianchi group. One describes the steps of the construction, different from the construction in [4], which permits us to determine the first known examples in M₂(Q (i√39)) and M₂(Q (i√6 respectively.     

The non-isolated vertices in the generating graph of a direct powers of simple groups

For a finite group G let Γ(G) denote the graph defined on the non-identity elements of G in such a way that two distinct vertices are connected by an edge if and only if they generate G. Many deep results on the generation of the finite simple groups G can be equivalently stated as theorems that ensure that Γ(G) is a rich graph, with several good properties. In this paper we want to consider Γ(G ^δ ) where G is a finite non-abelian simple group and G ^δ is the largest 2-generated power of G, with the aim to investigate whether the good generation properties of G still affect the behaviour of Γ(G ^δ ). In particular we prove that the graph obtained from Γ(G ^δ ) by removing the isolated vertices is 1-arc transitive and connected and we investigate the diameter of this graph. Moreover, some intriguing open questions will be introduced and their solutions will be exemplified for $G=\operatorname{Alt}(5)$ .     

Malnormal subgroups of lattices and the Pukánszky invariant in group factors

Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II₁ factor VN(Γ) contains a masa A with Pukánszky invariant {∞}.     

On regular graphs,V

Let Γ₃ be an infinite regular tree of valence 3. There exist subgroups B of Aut (Γ₃) which are 5-regular on Γ₃, i.e., sharply transitive on the set of 5-arcs of Γ₃. We prove that any two such subgroups are conjugate in Aut (Γ₃). The pair (Γ₃, B) is a universal 5-regular action in the sense that if (G, A) is a pair consisting of a cubical graph G and a 5-regular subgroup A of automorphisms of G then (G, A) can be “covered” by (Γ₃, B) in a certain natural way.     

On a question of Külshammer for representations of finite groups in reductive groups

Let G be a simple algebraic group of type G₂ over an algebraically closed field of characteristic 2. We give an example of a finite group Γ with Sylow 2-subgroup Γ₂ and an infinite family of pairwise non-conjugate homomorphisms ρ: Γ → G whose restrictions to Γ₂ are all conjugate. This answers a question of Burkhard Külshammer from 1995. We also give an action of Γ on a connected unipotent group V such that the map of 1-cohomologies H¹(Γ, V) → H¹(Γ_p, V) induced by restriction of 1-cocycles has an infinite fibre.     

Traces on Noncommutative Homogeneous Spaces

We study properties of C*-algebraic deformations of homogeneous spaces G/Γ which are equivariant in the sense that they preserve the natural action of G by left translation. The center is shown to be isomorphic to C(G/G⁰_ρ) for a certain subgroup G⁰_ρ of G, and there is a 1-1 correspondence between normalised traces and probability measures on G/G⁰_ρ. This makes it possible to represent the deformed algebra as operators over L²(G/Γ). Applications to K-theory are also mentioned.     

On a theorem of Rhemtulla on polycyclic groups

Let H be a subgroup of the polycyclic-by-finite group G and denote the automorphism group of G by Γ. We prove that there exists an integer d such that in the poset {?_γ∈∑H^γ:∑ a subset of Γ} of all intersections of images H^γ of H under Γ, chains have length at most d. In particular the poset satisfies the minimal condition. This extends and improves a theorem of A.H. Rhemtulla. We also provide a very different proof of Rhemtulla’s theorem.     

Groups with the same non-commuting graph

The non-commuting graph Γ_G of a non-abelian group G is defined as follows. The vertex set of Γ_G is G−Z(G) where Z(G) denotes the center of G and two vertices x and y are adjacent if and only if xy≠yx. It has been conjectured that if G and H are two non-abelian finite groups such that Γ_G≅Γ_H, then |G|=|H| and moreover in the case that H is a simple group this implies G≅H. In this paper, our aim is to prove the first part of the conjecture for all the finite non-abelian simple groups H. Then for certain simple groups H, we show that the graph isomorphism Γ_G≅Γ_H implies G≅H.     

Torsion and Abelianization in Equivariant Cohomology

Let X be a topological space upon which a compact connected Lie group G acts. It is well known that the equivariant cohomology H ^* _G (X; Q) is isomorphic to the subalgebra of Weyl group invariants of the equivariant cohomology H ^* _T (X; Q), where T is a maximal torus of G. This relationship breaks down for coefficient rings k other than Q. Instead, we prove that under a mild condition on k the algebra H ^* _G (X; k) is isomorphic to the subalgebra of H ^* _T (X; k) annihilated by the divided difference operators.     

Finiteness results for lattices in certain Lie groups

In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups G which possess certain “density” properties (see Moskowitz, M., On the density theorems of Borel and Furstenberg, Ark. Mat. 16 (1978), 11–27, and Moskowitz, M., Some results on automorphisms of bounded displacement and bounded cocycles, Monatsh. Math. 85 (1978), 323–336). For such groups we show that Γ always has finite index in its normalizer N _G (Γ). We then investigate analogous questions for the automorphism group Aut(G) proving, under appropriate conditions, that Stab_Aut(G)(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup \(\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}\), of Aut(G) has finite index in Stab_Aut(G)(Γ). We test the limits of our results with various examples and counterexamples.     

Invariant subspaces of some function spaces on a locally compact group

Let G be a σ-compact and locally compact group. If f?L^∞(G) let U_f be the closed subspace of L^∞(G) generated by the left translations of f. Conditions are given which ensure that each function in U_f may be expanded in an essentially unique way as an absolutely convergent series of translations of f. In this case U_f contains subspaces which are isometrically isomorphic to l¹. If G is metrizable and nondiscrete there is a continuum Γ in L^∞(G) such that, for each f?Γ, U_f contains no non-zero continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. If G is non-compact, metrizable, and non-discrete there is a continuum Γ of bounded continuous functions on G such that, for each f?Γ, U_f contains no non-zero left uniformly continuous function, and for f, g?Γ with f ≠ g, U_f ∩ U_g = {0}. The subspaces U_f above are translation invariant but are not convolution invariant.     

On approximation of Lie groups by discrete subgroups

A locally compact group G is said to be approximated by discrete subgroups (in the sense of Tôyama) if there is a sequence of discrete subgroups of G that converges to G in the Chabauty topology (or equivalently, in the Vietoris topology). The notion of approximation of Lie groups by discrete subgroups was introduced by Tôyama in Kodai Math. Sem. Rep. 1 (1949) 36–37 and investigated in detail by Kuranishi in Nagoya Math. J. 2 (1951) 63–71. It is known as a theorem of Tôyama that any connected Lie group approximated by discrete subgroups is nilpotent. The converse, in general, does not hold. For example, a connected simply connected nilpotent Lie group is approximated by discrete subgroups if and only if G has a rational structure. On the other hand, if Γ is a discrete uniform subgroup of a connected, simply connected nilpotent Lie group G then G is approximated by discrete subgroups Γ _n containing Γ. The proof of the above result is by induction on the dimension of G, and gives an algorithm for inductively determining Γ _n . The purpose of this paper is to give another proof in which we present an explicit formula for the sequence (Γ _n ) _n?≥?0 in terms of Γ. Several applications are given.     

Tetravalent Edge-transitive Cayley Graphs of PGL （2, p）

t Let F = Cay（G, S）, R（G） be the right regular representation of G. The graph Г is called normal with respect to G, if R（G） is normal in the full automorphism group Aut（F） of F. Г is called a bi-normal with respect to G if R（G） is not normal in Aut（Г）, but R（G） contains a subgroup of index 2 which is normal in Aut（F）. In this paper, we prove that connected tetravalent edge-transitive Cayley graphs on PGL（2,p） are either normal or bi-normal when p ≠ 11 is a prime.     

Can one hear the shape of a group?

Let Γ be a finitely generated group, and letS be a finite, non-necessarily symmetric, generating subset of Γ. Leth be the transition operator of the directed Cayley graphG(Γ,S), acting onl ² (Γ). Staring with Kesten’s seminal results, we give a survey of results linking group-theoretic properties of the pair (Γ,S) with spectral properties ofh.     

Integral formulas with distribution kernels for irreducible projections in L2 of a nilmanifold

Let N be a simply connected nilpotent Lie group and Γ a discrete uniform subgroup. The authors consider irreducible representations σ in the spectrum of the quasi-regular representation N × L²(Γ/N) → L²(Γ→) which are induced from normal maximal subordinate subgroups M ? N. The primary projection P_σ and all irreducible projections P ? P_σ are given by convolutions involving right Γ-invariant distributions D on Γ→, $\text{Pf(Γn) = D 1 f(Γn) = <D, n · f>}\text{all}\text{f ? C}{}^{\mathrm{\infty }}\text{(Γ/N)}$, where n · f(ζ) = f(ζ · n). Extending earlier work of Auslander and Brezin, and L. Richardson, the authors give explicit character formulas for the distributions, interpreting them as sums of characters on the torus T^κ = (Γ ∩ M) · [M, M]?M. By examining these structural formulas, they obtain fairly sharp estimates on the order of the distributions: if σ is associated with an orbit $\text{O}$ ? $\text{n}$¹ and if $\text{V}$ ? $\text{n}$¹ is the largest subspace which saturates θ in the sense that f ? $\text{O}$ ? f + $\text{V}$ ? $\text{O}$. As a corollary they obtain Richardson's criterion for a projection to map C⁰(Γ→) into itself. The authors also resolve a conjecture of Brezin, proving a Zero-One law which says, among other things, that if the primary projection P_σ maps C^r(Γ→) into C⁰(Γ→), so do all irreducible projections P ? P_σ. This proof is based on a classical lemma on the extent to which integral points on a polynomial graph in Rⁿ lie in the coset ring of Zⁿ (the finitely additive Boolean algebra generated by cosets of subgroups in Zⁿ). This lemma may be useful in other investigations of nilmanifolds.     

Equivariant gerbes over compact simple Lie groups

Using groupoid S¹-central extensions, we present, for a compact simple Lie group G, an infinite dimensional model of S¹-gerbe over the differential stack G/G whose Dixmier–Douady class corresponds to the canonical generator of the equivariant cohomology H_G³(G). To cite this article: K. Behrend et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Grundy number and products of graphs

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedyk-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertices of G. In this paper, we study the Grundy number of the lexicographic and cartesian products of two graphs in terms of the Grundy numbers of these graphs.Regarding the lexicographic product, we show that Γ(G)×Γ(H)≤Γ(G[H])≤2^Γ(G)−1(Γ(H)−1)+Γ(G). In addition, we show that if G is a tree or Γ(G)=Δ(G)+1, then Γ(G[H])=Γ(G)×Γ(H). We then deduce that for every fixed c≥1, given a graph G, it is CoNP-Complete to decide if Γ(G)≤c×χ(G) and it is CoNP-Complete to decide if Γ(G)≤c×ω(G).Regarding the cartesian product, we show that there is no upper bound of Γ(G□H) as a function of Γ(G) and Γ(H). Nevertheless, we prove that Γ(G□H)≤Δ(G)⋅2^Γ(H)−1+Γ(H).